The probability generating functions show that Sn follows a negative binomial distribution with parameters n and β. Expanding the generating function, we find that Gn(z) = E(z^Sn) = E(z^(N1+...+Nn)) = E(z^N1... z^Nn). The probability that Sn takes values less than 40 is approximately 0.0012. The probability that Sn is less than 40 is approximately 0.0012.
(a) By using the probability generating functions, show that Sn follows a negative binomial distribution.
Using probability generating functions, the generating function of Ni is given by:
G(z) = E(z^Ni) = Σ(z^ni * P(Ni=ni)),
where P(Ni=ni) = (1−β)^(ni−1) * β (for ni=1,2,3,...).
Therefore, the generating function of Sn is:
Gn(z) = E(z^Sn) = E(z^(N1+...+Nn)) = E(z^N1 ... z^Nn).
From independence, we have:
Gn(z) = G(z)^n = (β/(1−(1−β)z))^n.
Now we need to expand the generating function Gn(z) using the Binomial Theorem:
Gn(z) = (β/(1−(1−β)z))^n = β^n * (1−(1−β)z)^−n = Σ[k=0 to infinity] (β^n) * ((−1)^k) * binomial(−n,k) * (1−β)^k * z^k.
Therefore, Sn has a Negative Binomial distribution with parameters n and β.
(b) With n=50 and β=2, find Pr[Sn < 40] by (i) the exact distribution and by (ii) the normal approximation.
(i) Using the exact distribution:
The probability that Sn takes values less than 40 is:
Pr(S50<40) = Σ[k=0 to 39] (50+k−1 k) * (2/(2+1))^k * (1/3)^(50) ≈ 0.001340021.
(ii) Using the normal approximation:
The mean of Sn is μ = 50 * 2 = 100, and the variance of Sn is σ^2 = 50 * 2 * (1+2) = 300.
Therefore, Sn can be approximated by a Normal distribution with mean μ and variance σ^2:
Sn ~ N(100, 300).
We can standardize the value 40 using the normal distribution:
Z = (Sn − μ) / σ = (40 − 100) / √(300/50) = -3.08.
Using the standard normal distribution table, we find:
Pr(Sn<40) ≈ Pr(Z<−3.08) ≈ 0.0012.
So the probability that Sn is less than 40 is approximately 0.0012.
To know more about binomial distribution Visit:
https://brainly.com/question/29163389
#SPJ11
During a sale a store offered 70% discount on a particular camera that was originally price at $450 what was the price of the camera after the discount
Answer:
$135
Step-by-step explanation:
Assuming that the 70% discount was applied on the original price of $450,
you multiply 0.7 (70%) to 450 and subtract that value from the original price. Basically, you are left with 30% of the original price.
Imagine taking the full price of the camera and subtracting 70% off that original price.
450 - (0.7)(450) = $135
The price of the camera after the 70% off discount is $135.
The height of a triangle is 8ft less than the base x. The area is 120ft2. Part: 0/3 Part 1 of 3 (a) Write an equation in tes of x that represents the given relationship. The equation is
The required equation in terms of x that represents the given relationship is x² - 8x - 240 = 0.
Given that the height of a triangle is 8ft less than the base x. Also, the area is 120ft². We need to find the equation in terms of x that represents the given relationship of the triangle. Let's solve it.
Step 1: We know that the formula to calculate the area of a triangle is, A = 1/2 × b × h, Where A is the area, b is the base, and h is the height of the triangle.
Step 2: The height of a triangle is 8ft less than the base x. So, the height of the triangle is x - 8 ft.
Step 3: The area of the triangle is given as 120 ft².So, we can write the equation as, A = 1/2 × b × hx - 8 = Height of the triangle, Base of the triangle = x, Area of the triangle = 120ft². Now substitute the given values in the formula to get an equation in terms of x.120 = 1/2 × x × (x - 8)2 × 120 = x × (x - 8)240 = x² - 8xSo, the equation in terms of x that represents the given relationship isx² - 8x - 240 = 0.
Let's learn more about triangle:
https://brainly.com/question/1058720
#SPJ11
maximum size of logical address space supported by this system is 1MB. a) How many frames are there in this system? 4096
2,147,483,648
=524288 frames or 2 31
/2 12
=2 19
=524288 frames b) What is the maximum number of frames that can be allocated to a process in this system? 4KB
1MB
= 2 12
2 20
=2 8
=256 c) How many bits are needed to represent the following: i. The page number 8 ii. The offset 12
a. there are 524,288 frames in the system. b. the maximum number of frames is determined by the number of bits required to represent the page number, and the number of pages that can be addressed is limited by the size of the logical address space. c. 3 bits are needed to represent the page number 8, and 12 bits are needed to represent the offset 12.
a) The system has 524,288 frames. This can be calculated by dividing the maximum size of the logical address space (1MB) by the size of each frame (4KB).
1MB = 2^20 bytes
4KB = 2^12 bytes
Number of frames = (1MB / 4KB) = (2^20 / 2^12) = 2^(20-12) = 2^8 = 256
Therefore, there are 524,288 frames in the system.
b) The maximum number of frames that can be allocated to a process in this system is 256. This is because the maximum number of frames is determined by the number of bits required to represent the page number, and the number of pages that can be addressed is limited by the size of the logical address space.
c) i. The page number 8 can be represented using 3 bits. This is because there are 2^3 = 8 possible page numbers (0 to 7).
ii. The offset 12 can be represented using 12 bits. This is because there are 2^12 = 4,096 possible offsets (0 to 4,095).
Therefore, 3 bits are needed to represent the page number 8, and 12 bits are needed to represent the offset 12.
Learn more about frames here
https://brainly.com/question/29649442
#SPJ11
Given that the value in 'total' is 564 and the value in 'answer' is 318096 , what will be the output from the following line? WriteLine(" {0} squared is {1:N0}", total, answer); 564 squared is 318,096.00 564 squared is 318,096 564 squared is 318096 564.00 squared is 318,096 No answer text provided. 564 squared is 318,096.0
The output from the given line of code, WriteLine(" {0} squared is {1:N0}", total, answer), will be "564 squared is 318,096".
The "{0}" placeholder is replaced with the value of 'total' (which is 564), and the "{1:N0}" placeholder is replaced with the value of 'answer' (which is 318,096) formatted with thousands separators.
The ":N0" format specifier ensures that the number is displayed with no decimal places and with thousands separators.
Therefore, the output will be a formatted string stating "564 squared is 318,096", where the number 318,096 is displayed with a comma separator for thousands.
The concept involves using the WriteLine function in programming to display formatted output. In this specific case, the line "WriteLine(" {0} squared is {1:N0}", total, answer);" uses placeholders {0} and {1} to insert the values of 'total' and 'answer' respectively. The ":N0" format specifier is used to display 'answer' with thousand separators. As a result, the output will display the message "564 squared is 318,096.00" with the appropriate values and formatting.
To know more about code refer to-
https://brainly.com/question/17204194
#SPJ11
A batting average in baseball is a statistical probability that measures a batter’s success at the plate. Is this an example of a binomial probability? If so, how? If not, why not? Think about the following: How are success and failure determined? How is a trial defined? Is each at bat a player makes repeatable and independent? How does a batting average answer the question of what is the probability of r successes in n trials?
Binomial probability is a statistical concept describing the likelihood of a binomial event occurring in baseball. Success is determined by a hit, while failure is an out. A batting average calculates the probability of hitting in one at-bat, but does not directly answer the question of r successes in n trials.
Binomial probability is a term used in statistics and probability to denote the likelihood of a binomial event occurring. The question about whether a batting average in baseball is a binomial probability is not a straightforward yes or no. However, this can be explained by considering the following:
In baseball, success is defined as a hit and failure is defined as an out. A hit is when a player strikes the ball and reaches base without being thrown out. An out is when a player strikes the ball and is thrown out before reaching base.
Each time a batter goes up to bat, it is considered a trial.
Each at-bat a player makes is independent because it is not affected by the previous at-bat or the next at-bat. For example, if a batter hits a home run, it does not increase the probability of hitting a home run in the next at-bat.
Batting average is defined as the number of hits a player gets divided by the number of at-bats. Therefore, it answers the question of what is the probability of getting a hit in one at-bat. For example, if a player has a batting average of 0.300, it means that they get a hit 30% of the time they go up to bat. However, it does not directly answer the question of what is the probability of r successes in n trials because each at-bat is independent. Therefore, to answer that question, we would need to use binomial probability.
To know more about Binomial probability Visit:
https://brainly.com/question/12474772
#SPJ11
Given a 32×8ROM chip with an enable input, show the external connections necessary to construct a 128×8ROM with four chips and a decoder.
The combination of the decoder and the 32×8ROM chips forms a 128×8ROM memory system.
To construct a 128×8ROM with four 32×8ROM chips and a decoder, the following external connections are necessary:
Step 1: Connect the enable inputs of all the four 32×8ROM chips to the output of the decoder.
Step 2: Connect the output pins of each chip to the output pins of the next consecutive chip. For instance, connect the output pins of the first chip to the input pins of the second chip, and so on.
Step 3: Ensure that the decoder has 2 select lines, which are used to select one of the four chips. Connect the two select lines of the decoder to the two highest-order address bits of the four 32×8ROM chips. This connection will enable the decoder to activate one of the four chips at a time.
Step 4: Connect the lowest-order address bits of the four 32×8ROM chips directly to the lowest-order address bits of the system, such that the address lines A0-A4 connect to each of the four chips. The highest-order address bits are connected to the decoder.Selecting a specific chip by the decoder enables the chip to access the required memory locations.
Thus, the combination of the decoder and the 32×8ROM chips forms a 128×8ROM memory system.
Know more about memory system:
https://brainly.com/question/28167719
#SPJ11
Paulina has $415, and she's mowing lawns to earn more money. If she charges $12 for each lawn, how much money, M, will she have after mowing L lawns? Select an equation that could be used to answer the question.
The equation that could be used to answer the question is: M = $12L + $415.
To solve the problem using an equation, let the total amount of money she makes after mowing L lawns be represented by M, and the number of lawns mowed be represented by L.
Since Paulina charges $12 for each lawn, the amount of money she earns from mowing one lawn is $12.
Since she's mowing L lawns, then the total amount of money she earns is $12L. If she has $415 initially, then her total amount of money after mowing L lawns will be the sum of the initial amount of money she had and the amount of money she earned from mowing L lawns.
Therefore, the equation that could be used to answer the question is: M = $12L + $415.
Know more about equation here:
https://brainly.com/question/29174899
#SPJ11
Determine which of the following statements are true.
a)There exists a natural number x such that x2-x=0.
b)For every natural number x, we find +1≥2.
c)For every real number x, we have √2 = x.
d)There exists a natural number x such that x²+5x+6=0.
Based on the given data, statement (a) is true, statement (b) is false and statement (c)is false and statement (d) is true.
The statement is true. There exists a natural number x such that x² - x = 0. By factoring the equation, we have x(x - 1) = 0. Therefore, the solutions are x = 0 and x = 1, both of which are natural numbers.
The statement is false. For every natural number x, we cannot have x + 1 ≥ 2. This inequality implies that every natural number x is greater than or equal to 1, which is not true since natural numbers start from 1 and are greater than 1.
The statement is false. For every real number x, we cannot have √2 = x. The square root of 2 (√2) is an irrational number, meaning it cannot be expressed as a fraction or a terminating or repeating decimal. Therefore, it is not possible for x to equal √2, as √2 is not a real number.
The statement is true. There exists a natural number x such that x² + 5x + 6 = 0. By factoring the quadratic equation, we have (x + 2)(x + 3) = 0. Therefore, the solutions are x = -2 and x = -3, both of which are natural numbers.
Learn more about natural number here:
brainly.com/question/32686617
#SPJ11
For the following exercise, solve the quadratic equation by factoring. 2x^(2)+3x-2=0
The solutions of the quadratic equation 2x^2 + 3x - 2 = 0 are x = 1/2 and x = -2.
To solve the quadratic equation 2x^2 + 3x - 2 = 0 by factoring, you need to find two numbers that multiply to -4 and add up to 3.
Using the fact that product of roots of a quadratic equation;
ax^2 + bx + c = 0 is given by (a.c) and sum of roots of the equation is given by (-b/a),you can find the two numbers you are looking for.
The two numbers are 4 and -1,which means that the quadratic can be factored as (2x - 1)(x + 2) = 0.
Using the zero product property, we can set each factor equal to zero and solve for x:
(2x - 1)(x + 2) = 0
2x - 1 = 0 or x + 2 = 0
2x = 1 or x = -2
x = 1/2 or x = -2.
Therefore, the solutions of the quadratic equation 2x^2 + 3x - 2 = 0 are x = 1/2 and x = -2.
To know more about quadratic equation click here:
https://brainly.com/question/30098550
#SPJ11
Given (10,4) and (x,-2), find x such that the distance between through two points is 10.
Therefore, the two possible values for x such that the distance between the points (10,4) and (x,-2) is 10 are x = 18 and x = 2.
To find the value of x such that the distance between the points (10,4) and (x,-2) is 10, we can use the distance formula. The distance formula is given by:
d = √((x2 - x1)² + (y2 - y1)²)
In this case, we are given (10,4) as one point, and we want to find x such that the distance between (10,4) and (x,-2) is 10.
Using the distance formula, we can plug in the given values:
10 = √((x - 10)² + (-2 - 4)²)
Simplifying the equation, we get:
100 = (x - 10)^² + (-6)²
Expanding the equation further:
100 = (x² - 20x + 100) + 36
Combining like terms:
100 = x² - 20x + 136
Rearranging the equation:
x² - 20x + 36 = 0
Now we can solve this quadratic equation to find the values of x. However, this quadratic equation doesn't factor nicely, so we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 1, b = -20, and c = 36. Plugging in these values, we get:
x = (-(-20) ± √((-20)² - 4(1)(36))) / (2(1))
Simplifying further:
x = (20 ± √(400 - 144)) / 2
x = (20 ± √256) / 2
x = (20 ± 16) / 2
This gives us two possible values for x:
x1 = (20 + 16) / 2 = 36 / 2 = 18
x2 = (20 - 16) / 2 = 4 / 2 = 2
Therefore, the two possible values for x such that the distance between the points (10,4) and (x,-2) is 10 are x = 18 and x = 2.
To know more about distance visit:
https://brainly.com/question/33716087
#SPJ11
Part C2 - Oxidation with Benedict's Solution Which of the two substances can be oxidized? What is the functional group for that substance? Write a balanced equation for the oxidation reaction with chr
Benedict's solution is commonly used to test for the presence of reducing sugars, such as glucose and fructose. In this test, Benedict's solution is mixed with the substance to be tested and heated. If a reducing sugar is present, it will undergo oxidation and reduce the copper(II) ions in Benedict's solution to copper(I) oxide, which precipitates as a red or orange precipitate.
To determine which of the two substances can be oxidized with Benedict's solution, we need to know the nature of the functional group present in each substance. Without this information, it is difficult to determine the substance's reactivity with Benedict's solution.
However, if we assume that both substances are monosaccharides, such as glucose and fructose, then they both contain an aldehyde functional group (CHO). In this case, both substances can be oxidized by Benedict's solution. The aldehyde group is oxidized to a carboxylic acid, resulting in the reduction of copper(II) ions to copper(I) oxide.
The balanced equation for the oxidation reaction of a monosaccharide with Benedict's solution can be represented as follows:
C₆H₁₂O₆ (monosaccharide) + 2Cu₂+ (Benedict's solution) + 5OH- (Benedict's solution) → Cu₂O (copper(I) oxide, precipitate) + C₆H₁₂O₇ (carboxylic acid) + H₂O
It is important to note that without specific information about the substances involved, this is a generalized explanation assuming they are monosaccharides. The reactivity with Benedict's solution may vary depending on the functional groups present in the actual substances.
To know more about Benedict's solution refer here:
https://brainly.com/question/12109037#
#SPJ11
How many ways exist to encage 5 animals in 11 cages if all of
them should be in different cages.
Answer:
This problem can be solved using the permutation formula, which is:
nPr = n! / (n - r)!
where n is the total number of items (cages in this case) and r is the number of items (animals in this case) that we want to select and arrange.
In this problem, we want to select and arrange 5 animals in 11 different cages, so we can use the permutation formula as follows:
11P5 = 11! / (11 - 5)!
= 11! / 6!
= 11 x 10 x 9 x 8 x 7
= 55,440
Therefore, there are 55,440 ways to encage 5 animals in 11 cages if all of them should be in different cages.
A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant, e.g., 1,3,9, … Write a method that checks if a given integer list (more than two elements) can be sorted into a geometric sequence, using the following header:
public static boolean canBeSortedGeoSeq(int[] list)
A. Please complete the following program:
1 public static boolean canBeSortedGeoSeq(int[] list) {
2 ______________________(list);
3 int ratio = list[1]/list[0];
4 int n = ___________;
5 for (int i=____; i
6 if ((list[_____]/list[_____])!=ratio)
7 return ______;
8 }
9 return ______;
10 }
B. If the list passed to the method is {2 4 3 6}, what will be the output from the code in A? To make the code work with double typed radio, how can we revise the code? For the same list {2 4 3 6}, what will be the output after the revision? What might be the new problem with the revised code?
The given program implements a method, canBeSortedGeoSeq, that checks if a given integer list can be sorted into a geometric sequence. The program sorts the list in ascending order and calculates the ratio between consecutive terms. It then iterates through the sorted list, comparing the ratio of each pair of consecutive terms with the initial ratio. If any ratio differs, the method returns false, indicating that the list cannot be sorted into a geometric sequence. Otherwise, it returns true.
A.
The complete program after filling the blanks is:
1 public static boolean canBeSortedGeoSeq(int[] list) {
2 Arrays.sort(list);
3 int ratio = list[1] / list[0];
4 int n = list.length;
5 for (int i = 1; i < n - 1; i++) {
6 if ((list[i + 1] / list[i]) != ratio)
7 return false;
8 }
9 return true;
10 }
B.
If the list passed to the method is {2, 4, 3, 6}, the output from the original code will be false. This is because the ratio between consecutive terms is not constant (2/4 = 0.5, 4/3 ≈ 1.33, 3/6 = 0.5).
To make the code work with double-typed ratio, we can revise the code by changing the data type of the ratio variable to double and modifying the comparison in the if statement accordingly:
public static boolean canBeSortedGeoSeq(int[] list) {
Arrays.sort(list);
double ratio = (double) list[1] / list[0];
int n = list.length;
for (int i = 1; i < n - 1; i++) {
if (((double) list[i + 1] / list[i]) != ratio)
return false;
}
return true;
}
After the revision, if the list passed is {2, 4, 3, 6}, the output will be false because the ratio is not constant (2/4 = 0.5, 4/3 ≈ 1.33, 3/6 = 0.5).
The new problem with the revised code is that it may encounter precision errors when performing division operations on floating-point numbers. Due to the limited precision of floating-point arithmetic, small differences in calculations can occur, leading to unexpected results.
In the case of checking geometric sequences, this can cause the program to mistakenly identify a non-geometric sequence as a geometric sequence or vice versa.
To address this issue, it is recommended to use a tolerance or epsilon value when comparing floating-point numbers to account for the precision limitations.
To learn more about geometric sequence: https://brainly.com/question/29632351
#SPJ11
Let X 1
,…,X n
∼Beta(θ,2). Show that T=∏ i=1
n
X i
is a sufficient statistic for θ. Note: You may simplify the pdf before you proceed f(x∣θ)= Γ(θ)Γ(2)
Γ(θ+2)
x θ−1
(1−x) 2−1
To show that the product statistic T = ∏ᵢ₌₁ⁿ Xᵢ is sufficient for θ, we need to demonstrate that the conditional distribution of the sample given T does not depend on θ.
Given that X₁, X₂, ..., Xₙ are i.i.d. random variables with a Beta distribution Beta(θ, 2), we can express the joint probability density function (pdf) of the sample as:
f(x₁, x₂, ..., xₙ | θ) = ∏ᵢ₌₁ⁿ f(xᵢ | θ)
= ∏ᵢ₌₁ⁿ [Γ(θ)Γ(2) / Γ(θ + 2)] * xᵢ^(θ - 1) * (1 - xᵢ)^(2 - 1)
= [Γ(θ)Γ(2) / Γ(θ + 2)]ⁿ * ∏ᵢ₌₁ⁿ xᵢ^(θ - 1) * (1 - xᵢ)
To proceed, let's rewrite the joint pdf in terms of the product statistic T:
f(x₁, x₂, ..., xₙ | θ) = [Γ(θ)Γ(2) / Γ(θ + 2)]ⁿ * T^(θ - 1) * (1 - T)^(2n - θ)
Now, let's factorize the joint pdf into two parts, one depending on the data and the other on the parameter:
f(x₁, x₂, ..., xₙ | θ) = g(T, θ) * h(x₁, x₂, ..., xₙ)
where g(T, θ) = [Γ(θ)Γ(2) / Γ(θ + 2)]ⁿ * T^(θ - 1) * (1 - T)^(2n - θ) and h(x₁, x₂, ..., xₙ) = 1.
The factorization shows that the joint pdf can be separated into a function of T, which depends on the parameter θ, and a function of the data x₁, x₂, ..., xₙ. Since the factorization does not depend on the specific values of x₁, x₂, ..., xₙ, we can conclude that the product statistic T = ∏ᵢ₌₁ⁿ Xᵢ is a sufficient statistic for θ.
To know more about Beta distribution, visit:
https://brainly.com/question/32657045
#SPJ11
A machine has four components, A, B, C, and D, set up in such a manner that all four parts must work for the machine to work properly. Assume the probability of one part working does not depend on the functionality of any of the other parts. Also assume that the probabilities of the individual parts working are P(A)=P(B)=0.95,P(C)=0.99, and P(D)=0.91. Find the probability that the machine works properly. Round to the nearest ten-thousandth. A) 0.8131 B) 0.8935 C) 0.1869 D) 0.8559
The probability of a machine functioning properly is P(A and B and C and D). The components' working is independent, so the probability is 0.8131. The correct option is A.
Given:P(A) = P(B) = 0.95P(C) = 0.99P(D) = 0.91The machine has four components, A, B, C, and D, set up in such a manner that all four parts must work for the machine to work properly.
Therefore,
The probability that the machine will work properly = P(A and B and C and D)
Probability that the machine works properly
P(A and B and C and D) = P(A) * P(B) * P(C) * P(D)[Since the components' working is independent of each other]
Substituting the values, we get:
P(A and B and C and D) = 0.95 * 0.95 * 0.99 * 0.91
= 0.7956105
≈ 0.8131
Hence, the probability that the machine works properly is 0.8131. Therefore, the correct option is A.
To know more about Probability Visit:
https://brainly.com/question/31828911
#SPJ11
Let f(n)=10log 10
(100n) and g(n)=log 2
n. Which holds: f(n)=O(g(n))
g(n)=O(f(n))
f(n)=O(g(n)) and g(n)=O(f(n))
After comparing the growth rates of f(n) and g(n) and observing the logarithmic function, we can say that f(n) = O(g(n)).
To determine which holds among the given options, let's compare the growth rates of f(n) and g(n).
First, let's analyze f(n):
f(n) = 10log10(100n)
= 10log10(10^2 * n)
= 10 * 2log10(n)
= 20log10(n)
Now, let's analyze g(n):
g(n) = log2(n)
Comparing the growth rates, we observe that g(n) is a logarithmic function, while f(n) is a with a coefficient of 20. Logarithmic functions grow at a slower rate compared to functions with larger coefficients.
Therefore, we can conclude that f(n) = O(g(n)), which means that option (a) holds: f(n) = O(g(n)).
To know more about logarithmic function, visit:
https://brainly.com/question/30339782#
#SPJ11
Here is a rectangle
the solution is in the attached figure
Suppose that the function f:[a,b]→R is integrable. Show that there exists an Archimedean sequence of partitions {Pn } for f on [a,b] such that P n+1 is a refinement of P n
for each n. Furthermore, for such a sequence of partitions, show that the sequence of upper Darboux sums is monotonically decreasing and the sequence of lower Darboux sums is monotonically increasing.
To prove the existence of an Archimedean sequence of partitions {Pn} for the integrable function f on [a, b], such that Pn+1 is a refinement of Pn for each n, we can use the concept of the Darboux sums.Let's consider an Archimedean number M, which means for any positive real number ε, there exists an integer N such that 1/N < ε. We can choose such an M that is greater than b - a, the length of the interval [a, b].
Now, let's construct a sequence of partitions {Pn} as follows: Divide the interval [a, b] into N subintervals of equal length, where N is a positive integer. Then, divide each of these subintervals into M subintervals of equal length. Repeat this process for each subsequent partition, resulting in finer and finer subdivisions.Since M is an Archimedean number, as N tends to infinity, the size of the subintervals tends to zero. Hence, the sequence of partitions {Pn} satisfies the condition that Pn+1 is a refinement of Pn for each n.Now, let's consider the sequence of upper Darboux sums and lower Darboux sums corresponding to the partitions {Pn}. As the partitions become finer, both the upper and lower Darboux sums converge to the definite integral of f over [a, b].Since each subsequent partition is a refinement of the previous partition, it follows that the upper Darboux sums are monotonically decreasing and the lower Darboux sums are monotonically increasing. This is because, with each refinement, the upper Darboux sum can only decrease as the suprema of the function over the subintervals become smaller, and the lower Darboux sum can only increase as the infima of the function over the subintervals become larger.
Learn more about Archimedean sequence here
https://brainly.com/question/32392329
#SPJ11
EQUATIONS AND INEQUALITIES Solving a word problem with three unknowns using a linear... The sum of three numbers is 105 . The second number is 4 times the third. The first number is 9 more than the th
The three numbers are:
x = 25
y = 64
z = 16
let x represent the first number, y represent the second number, and z represent the third number.
We can translate the given information into equations:
Equation 1: x + y + z = 105 (the sum of three numbers is 105).
Equation 2: y = 4z (the second number is 4 times the third).
Equation 3: x = z + 9 (the first number is 9 more than the third).
To solve this system of equations, we can substitute the expressions for y and x into Equation 1:
(z + 9) + (4z) + z = 105
Simplifying this equation, we get:
6z + 9 = 105
By subtracting 9 from both sides:
6z = 96
Dividing both sides by 6:
z = 16
Substituting the value of z into the other equations, we find:
y = 4z = 4 * 16 = 64
x = z + 9 = 16 + 9 = 25
Hence, the three numbers are 25, 64, and 16.
learn more about "equations":- https://brainly.com/question/29174899
#SPJ11
EQUATIONS AND INEQUALITIES Solving a word problem with three unknowns using a linear... The sum of three numbers is 105 . The second number is 4 times the third. The first number is 9 more than the third.
using 32-bit I-EEE-756 Format
1. find the smallest floating point number bigger than 230
2. how many floating point numbers are there between 2 and 8?
The smallest floating point number bigger than 2^30 in the 32-bit IEEE-756 format is 1.0000001192092896 × 2^30 and There are 2,147,483,648 floating point numbers between 2 and 8 in the same format.
1. In the 32-bit IEEE-756 format, the smallest floating point number bigger than 2^30 can be found by analyzing the bit representation. The sign bit is 0 for positive numbers, the exponent is 30 (biased exponent representation is used, so the actual exponent value is 30 - bias), and the fraction bits are all zeros since we want the smallest number. Therefore, the bit representation is 0 10011101 00000000000000000000000. Converting this back to decimal, we get 1.0000001192092896 × 2^30, which is the smallest floating point number bigger than 2^30.
2. To find the number of floating point numbers between 2 and 8 in the 32-bit IEEE-756 format, we need to consider the exponent range and the number of available fraction bits. In this format, the exponent can range from -126 to 127 (biased exponent), and the fraction bits provide a precision of 23 bits. We can count the number of unique combinations for the exponent (256 combinations) and multiply it by the number of possible fraction combinations (2^23). Thus, there are 256 * 2^23 = 2,147,483,648 floating point numbers between 2 and 8 in the given format.
Therefore, The smallest floating point number bigger than 2^30 in the 32-bit IEEE-756 format is 1.0000001192092896 × 2^30 and There are 2,147,483,648 floating point numbers between 2 and 8 in the same format.
To learn more about number click here
brainly.com/question/24908711
#SPJ11
Suppose we take a random sample of size from a continuous distribution having median 0 so that the probability of any one observation being positive is .5. We now disregard the signs of the observations, rank them from smallest to largest in absolute value, and then let the sum of the ranks of the observations having positive signs. For example, if the observations are , , and , then the ranks of positive observations are 2 and 3, so . In Chapter will be called Wilcoxon's signed-rank statistic. W can be represented as follows:
where the s are independent Bernoulli rv's, each with corresponds to the observation with rank being positive). Compute the following:
a. and then using the equation for [Hint: The first positive integers sum to b. and then [Hint: The sum of the squares of the first positive integers is
The value of Var(W) = n(n+1)(2n+1)/6.
Σ i² = n(n+1)(2n+1)/6.Σ i³ = (Σ i)² = (n(n+1)/2)² = (n²(n+1)²)/4.Σ [tex]i^4[/tex] = (n(n+1)(2n+1)(3n² + 3n - 1))/30.(a) W = Σ [tex]s_i[/tex] i,
where [tex]s_i[/tex] is an independent Bernoulli random variable with probability p = 0.5, indicating whether the observation with rank i is positive.
First, let's calculate E(W):
E(W) = E(Σ [tex]s_i[/tex] i)
= Σ E([tex]s_i[/tex] i) (linearity of expectation)
= Σ E([tex]s_i[/tex]) E(i) (independence)
= Σ 0.5 x i (E([tex]s_i[/tex]) = 0.5)
= 0.5 x Σ i
= 0.5 (1 + 2 + 3 + ... + n)
= 0.5 (n(n+1)/2)
= 0.25 n(n+1)
Next, let's calculate Var(W):
Var(W) = Var(Σ [tex]s_i[/tex] i)
= Σ Var([tex]s_i[/tex] i) + 2 Σ Σ Cov([tex]s_i[/tex] i, [tex]s_j[/tex] j)
= Σ Var([tex]s_i[/tex]) E(i)² + 2 Σ Σ Cov([tex]s_i[/tex] i, [tex]s_j[/tex] j)
= Σ (0.5 i²) + 2 Σ Σ Cov([tex]s_i[/tex] i, [tex]s_j[/tex] j)
= 0.5 Σ i² + 2 Σ Σ Cov([tex]s_i[/tex] i, [tex]s_j[/tex] j)
To calculate Cov([tex]s_i[/tex] i, [tex]s_i[/tex] j),
- When i ≠ j:
Cov([tex]s_i[/tex] i, [tex]s_i[/tex] j) = E([tex]s_i[/tex] i[tex]s_j[/tex] j) - E[tex]s_j[/tex] * i) * E([tex]s_j[/tex] j)
= E([tex]s_j[/tex]) E(i) E([tex]s_j[/tex]) E(j) - E([tex]s_i[/tex] i) E([tex]s_j[/tex] j)
= 0.5 i x 0.5 j - 0.5 i² 0.5 j²
= 0.25 i j - 0.25 i² j²
- When i = j:
Cov(s_i * i, s_i * i) = E(([tex]s_i[/tex] i)²) - E([tex]s_i[/tex] i)²
= E([tex]s_i[/tex]^2 i²) - E([tex]s_i[/tex] i)²
= E([tex]s_i[/tex]) * E(i²) - E([tex]s_i[/tex] i)²
= 0.5 i² - 0.5 i² × 0.5 i²
= 0.25 i²
Now, let's substitute these values back into the expression for Var(W):
Var(W) = 0.5 Σ i² + 2 Σ Σ Cov([tex]s_i[/tex] * i, [tex]s_j[/tex] * j)
= 0.5 Σ i² + 2 Σ Σ (0.25 *i j - 0.25 i² j²) (i ≠ j)
+ 2 Σ (0.25 i²) (i = j)
= 0.5 Σ i^2 + 2 Σ (0.25 i²)+ 2 Σ Σ (0.25 i j - 0.25 i² j²) (i ≠ j)
Using the hint provided, we can simplify the expression:
Σ i = n(n+1)/2,
Σ i² = n(n+1)(2n+1)/6,
Σ (i j) = n(n+1)(2n+1)/6,
Substituting these values back into the expression for Var(W):
Var(W) = 0.5 n(n+1)(2n+1)/6 + 2 (0.25 n(n+1)(2n+1)/6)
+ 2 (0.25 n(n+1)(2n+1)/6 - 0.25 n(n+1)(2n+1)/6) (i ≠ j)
= n(n+1)(2n+1)/12 + 0.5 n(n+1)(2n+1)/6
= n(n+1)(2n+1)(1/12 + 1/12)
= n(n+1)(2n+1)/6
(b) We are asked to compute Σ i².
Σ i² = n(n+1)(2n+1)/6.
(c) Using the hint provided, we can calculate Σ i³ as follows:
Σ i³ = (Σ i)² = (n(n+1)/2)² = (n²(n+1)²)/4.
(d) We are asked to compute Σ [tex]i^4[/tex].
Using the hint provided, we can calculate Σ[tex]i^4[/tex] as follows:
Σ [tex]i^4[/tex] = (n(n+1)(2n+1)(3n² + 3n - 1))/30.
Learn more about Bernoulli random variable here:
https://brainly.com/question/31143222
#SPJ4
For any random variable X and function g:supp(X)→R, the expectation of g(X) is E[g(X)]=∫ x∈supp(X)
g(x)f X
(x)dx 1. Let a and b be constants and X be a random variable. Show that E[a+bX]=a+bE[Y]. 2. We saw in class that variance for random variable X is defined as Var[X]≡E[(X−E[X]) 2
] Using your previous result, show that Var[X]=E[X 2
]−E[X] 2
3. Show that Var[a+bX]=b 2
Var[X]. 4. In the previous question, a does not contribute to the variance but b does. Why is this? An intuitive answer is enough. h(X,Y) is E[h(X,Y)]=∫ y∈supp(Y)
∫ x∈supp(X)
h(x,y)f X,Y
(x,y)dxdy The order of integration does not matter. In the equation above, we integrated out X first, but you could have integrated out Y first instead. 5. To see the linearity of expectations in full display, let a and b be constants, and let X and Y be random variables. Show that E[aX+bY]=aE[X]+bE[Y]. Hint: The expectation on the LHS involves both X and Y, so it involves the joint distribution of (X,Y). The expectations on the RHS involve either X or Y but not both, so they involve only the marginal distributions of X and Y. How do you get from joint distributions to marginal distributions? And remember you can switch the order of integration.
The given expectation formula is E[g(X)]=∫ x∈supp(X) g(x)fX(x)dx. Let a and b be constants and X be a random variable.
We have to show that E[a+bX]=a+bE[X].
We know that E[a + bX] = E[a] + E[bX]
Therefore, E[a + bX] = a + bE[X].
The variance of the random variable X is Var[X]≡E[(X−E[X])2].
Using the result of the previous question, we have to show that Var[X]=E[X2]−E[X]2.
The calculation is as follows:
Var[X] = E[(X - E[X])2] = E[X2 - 2XE[X] + E[X]2] = E[X2] - 2E[X]E[X] + E[X]2 = E[X2] - E[X]2
We have to show that Var[a+bX] = b2Var[X].
Using the result of the previous question, we get:
Var[a + bX] = E[(a + bX)2] - E[a + bX]2Var[a + bX] = E[a2 + 2abX + b2X2] - (a + bE[X])2
Var[a + bX] = a2 + 2abE[X] + b2E[X2] - (a2 + 2abE[X] + b2E[X]2)
Var[a + bX] = b2E[X2] - b2E[X]2
Var[a + bX] = b2Var[X]4.
In the previous question, a does not contribute to the variance but b does because the variance is a measure of the spread of the random variable around its mean, and the constant a does not affect the spread of the random variable, but b does. It scales the random variable's values and, therefore, affects its spread.5.
We need to show that E[aX+bY]=aE[X]+bE[Y], where X and Y are random variables and a and b are constants.
We know that E[aX + bY] = ∫∫[aX + bY] fXY(x, y) dxdy
= ∫∫aX fXY(x, y) dxdy + ∫∫bY fXY(x, y) dxdy
= a ∫∫X fXY(x, y) dxdy + b ∫∫Y fXY(x, y) dxdy
= a E[X] + b E[Y].
Therefore, we can see that the order of integration does not matter, and we can integrate out Y first or X first. The linearity of expectations comes from the linearity of integration.
To know more about random variable visit:
brainly.com/question/32680168
#SPJ11
they've identified mr.bottle snaps a potential candidate to claim and start mining. the probability Of funding gold at mt.bottlesnaap is 20%, silver is 50%, and a dragon is 30%
if a local insurance salesperson offered to sell the dwarves an insurance policy that would pay the dragon it's non-eating money should the mountain be home to a dragon (thus not allowing the dwarves to havr to pay 80 gold if there's a dragon there) what is the most that the band of dwarves could pay and still break even on the insurance policy given the expected cost?
The most that the band of dwarves could pay for the insurance policy and still break even is 40 gold.
To determine the most that the band of dwarves could pay and still break even on the insurance policy, we need to compare the expected cost without insurance to the cost with insurance.
Given the probabilities of finding gold, silver, and a dragon at Mt. Bottlesnaap:
Probability of finding gold (G) = 20%
Probability of finding silver (S) = 50%
Probability of finding a dragon (D) = 30%
Let's assume the expected cost without insurance is calculated as follows:
Expected cost without insurance = Cost(G) * P(G) + Cost(S) * P(S) + Cost(D) * P(D)
Assuming the cost of finding gold is 80 gold, and the cost of finding silver is 0 gold (no cost), and the cost of finding a dragon is 80 gold (to be paid without insurance), we can calculate the expected cost without insurance:
Expected cost without insurance = 80 * 0.2 + 0 * 0.5 + 80 * 0.3 = 16 + 0 + 24 = 40 gold
To break even on the insurance policy, the expected cost with insurance should be equal to the expected cost without insurance. Since the insurance policy pays the dragon its non-eating money (80 gold), the band of dwarves would not have to pay the cost if a dragon is found.
Therefore, the most that the band of dwarves could pay for the insurance policy and still break even is 40 gold.
Learn more about insurance policy from
https://brainly.com/question/1171936
#SPJ11
The following table categorizes a sample of overweight people based on their ages and genders. If a person is selected randomly from the sample, what is the probability that the selected person is a kid, knowing that the person is a female? 0.062 0.53 0.25 0.33 The following table categorizes a sample of overweight people based on their ages and genders. If a person is selected randomly from the sample, what is the probability that the selected person is either a male or a kid? 0.71 0.46 0.13 0.61 The following table categorizes a sample of overweight people based on their ages and genders. If a person is selected randomly from the sample, what is the probability that the selected person is a female, given that the person is a kid? 0.248 0.28 (C) 0.33 0.72 The following table categorizes a sample of overweight people based on their ages and genders. If a person is selected randomly from the sample, what is the probability that the selected person is a teenager? 0.038 (B) 0.082 0.12 0.68
The table categorizes overweight individuals based on age and gender. The probability of selecting a person as a kid is 0.47, while the probability of selecting a person as either a male or a kid is 0.77. The probability of selecting a person as a female is 0.55, and the probability of selecting a teenager is 0.038.So, correct option is A
The given table categorizes a sample of overweight people based on their ages and genders. We need to find the probability for the given conditions.
The solution for each question is as follows:
1. If a person is selected randomly from the sample, what is the probability that the selected person is a kid, knowing that the person is a female?The probability that the selected person is a kid given the person is a female will be the ratio of the number of females who are kids to the total number of females. From the given table, we can see that there are 35 + 31 = 66 females and out of those, 31 are kids. Therefore, the required probability is 31/66 = 0.47, which is closest to 0.5. Hence, the answer is not given in the options provided.2. If a person is selected randomly from the sample.
The probability that the selected person is either a male or a kid will be the ratio of the number of males plus the number of kids to the total number of people. From the given table, we can see that there are 36 + 31 + 25 + 28 = 120 males and kids. Therefore, the required probability is 92/120 = 0.77, which is closest to 0.8. Hence, the answer is not given in the options provided.3. If a person is selected randomly from the sample, what is the probability that the selected person is a female, given that the person is a kid?The probability that the selected person is a female given the person is a kid will be the ratio of the number of female kids to the total number of kids. From the given table, we can see that there are 31 female kids. Therefore, the required probability is
31/(25+31)
= 31/56
= 0.55
which is closest to 0.6. Hence, the answer is not given in the options provided.4. If a person is selected randomly from the sample,
The probability that the selected person is a teenager will be the ratio of the number of teenagers to the total number of people. From the given table, we can see that there are 8 teenagers. Therefore, the required probability is 8/210 = 0.038. Hence, the answer is option (A) 0.038.
To know more about probability Visit:
https://brainly.com/question/31828911
#SPJ11
Fill in the blank. The ________ is the probability of getting a test statistic at least as extreme as the one representing the sample data, assuming that the null hypothesis is true.
A. p-value
B. Critical value
C. Level of significance
D. Sample proportion
The p-value is the probability of getting a test statistic at least as extreme as the one representing the sample data, assuming that the null hypothesis is true.
The p-value is the probability of obtaining a test statistic that is as extreme as, or more extreme than, the one observed from the sample data, assuming that the null hypothesis is true. It is a measure of the evidence against the null hypothesis provided by the data. The p-value is used in hypothesis testing to make decisions about the null hypothesis. If the p-value is less than the predetermined level of significance (alpha), typically 0.05, it suggests that the observed data is unlikely to occur by chance alone under the null hypothesis. This leads to rejecting the null hypothesis in favor of the alternative hypothesis. On the other hand, if the p-value is greater than the significance level, there is insufficient evidence to reject the null hypothesis.For more questions on probability :
https://brainly.com/question/13786078
#SPJ8
suppose that the effects of copper on a second species of fish (say, species b) show the variance of ln(lc50) measurements to be 0.2. if the population means of ln(lc50) for the two species are equal, find the probability that, with random samples of seven measurements from each species, the sample mean for species a exceeds the sample mean for species b by at least 1 unit. (round your answer to four decimal places.)
Probability that, with random samples of seven measurements from each species, the sample mean for species A exceeds the sample mean for species B by at least 1 unit is 0.0019 .
Here,
Suppose that X and Y are two independent random samples. The variable X is normally distributed with mean µ1 and variance 0.4 and the variable Y is normally distributed with mean µ2 and variance 0.8 and each sample size is 10.
Now the X is also a random variable and it follows normal with mean µ1 and variance 0.4/10, i.e. 0.04
And Y is also random variable and it follows normal with mean µ2 and variance 0.8/10, i.e. 0.08
Now,
V[X - Y] = V[X] + V[Y] (Since the samples are independent.)
V[X - Y] = 0.04 + 0.08
= 0.12
Now, we need to find the probability that, the sample mean for species A exceeds the sample mean for species B by at least 1 unit.
∴P(X - Y ≥ 1)
So,
P(X - Y ≥ 1) = P[Z ≥ 1-0/0.12]
P(X - Y ≥ 1) = P[Z ≥ 2.8858]
= 1 - P[Z < 2.8858]
Therefore, the required probability,
= 0.0019
Know more about variance,
https://brainly.com/question/16686665
#SPJ4
a radar complex consists of 10 units that operate independently. the probability that a unit detects an incoming missile is 0.85. find the probability that an incoming missile will: (a) not be detected by any unit. (b) be detected by at least 8 units. (c) next year the radar complex will be expanded to 400 units. what will be the approximate probability that at least 360 units will detect an incoming missile.
Using binomial probability to solve the probability of the independent events;
(a) The probability that an incoming missile will not be detected by any unit in the radar complex is approximately 0.0000341468.
(b) The probability that an incoming missile will be detected by at least 8 units in the radar complex is approximately 0.999718.
(c) If the radar complex is expanded to 400 units with the same detection probability (0.85), the approximate probability that at least 360 units will detect an incoming missile is approximately 0.0265.
What is the probability that the incoming missile will not be detected by any unit?To solve these probability problems, we'll need to apply the concepts of independent events and the binomial probability formula. Let's go step by step:
(a) The probability that a unit does not detect an incoming missile is 1 - 0.85 = 0.15. Since each unit operates independently, the probability that none of the 10 units detects the missile is the product of their individual probabilities:
P(not detected by any unit) = (0.15)^10 = 0.0000341468 (approximately)
(b) To find the probability that an incoming missile is detected by at least 8 units, we need to calculate the probability of it being detected by exactly 8, exactly 9, or exactly 10 units, and then sum those probabilities.
P(detected by at least 8 units) = P(detected by 8 units) + P(detected by 9 units) + P(detected by 10 units)
Using the binomial probability formula:
P(k successes in n trials) = C(n, k) * p^k * (1-p)^(n-k)
where C(n, k) represents the number of combinations of n items taken k at a time, p is the probability of success, and (1-p) is the probability of failure.
P(detected by 8 units) = C(10, 8) * (0.85)^8 * (0.15)^2 ≈ 0.286476
P(detected by 9 units) = C(10, 9) * (0.85)^9 * (0.15)^1 ≈ 0.369537
P(detected by 10 units) = C(10, 10) * (0.85)^10 * (0.15)^0 = 0.443705
Summing these probabilities, we get:
P(detected by at least 8 units) ≈ 0.286476 + 0.369537 + 0.443705 ≈ 0.999718
Therefore, the probability that an incoming missile will be detected by at least 8 units is approximately 0.999718.
(c) If the radar complex is expanded to 400 units and the probability of detection remains the same (0.85), we can approximate the probability that at least 360 units will detect an incoming missile using a normal approximation to the binomial distribution.
The mean (μ) of the binomial distribution is given by n * p, and the standard deviation (σ) is given by √(n * p * (1-p)). In this case, n = 400 and p = 0.85.
μ = 400 * 0.85 = 340
σ = √(400 * 0.85 * 0.15) ≈ 10.2469
To find the probability that at least 360 units will detect an incoming missile, we can use the cumulative distribution function (CDF) of the normal distribution.
P(X ≥ 360) ≈ P(Z ≥ (360 - μ) / σ)
P(Z ≥ (360 - 340) / 10.2469) ≈ P(Z ≥ 1.951)
Consulting a standard normal distribution table or using a calculator, we find that P(Z ≥ 1.951) ≈ 0.0265.
Therefore, the approximate probability that at least 360 units will detect an incoming missile with the expanded radar complex is approximately 0.0265.
Learn more on binomial probability here;
https://brainly.com/question/15246027
#SPJ4
Julie's family is filling up the pool in her backyard. The equation y=8,400+5. 2x can be used to show the rate of which the pool is filling up
a) Julie's pool is filling at a faster rate than Elaina's pool.
b) Julie's pool initially contained more water than Elaina's pool.
c) After 30 minutes, Julie's pool will contain more water than Elaina's pool.
a. To determine which pool is filling at a faster rate, we can compare the values of the rate of filling for Julie's pool and Elaina's pool at any given time.
Let's calculate the rates of filling for both pools using the provided equation.
For Julie's pool:
y = 8,400 + 5.2x
Rate of filling is 5.2 gallons per minute.
For Elaina's pool:
At t = 0 minutes, the pool contained 7,850 gallons.
At t = 3 minutes, the pool contained 7,864.4 gallons.
Rate of filling for Elaina's pool from t = 0 to t = 3:
= (7,864.4 - 7,850) / (3 - 0)
= 14.4 / 3
= 4.8 gallons per minute.
Rate of filling is 4.8 gallons per minute.
As 5.2>4.8. So, Julie's pool is filling up at a faster rate than Elaina's pool, which remains constant at 4.8 gallons per minute.
b. To determine which pool initially contained more water, we need to evaluate the number of gallons in each pool at t = 0 minutes.
For Julie's pool: y = 8,400 + 5.2(0) = 8,400 gallons initially.
Elaina's pool contained 7,850 gallons initially.
Therefore, Julie's pool initially contained more water than Elaina's pool.
c. To determine which pool will contain more water after 30 minutes, we can substitute x = 30 into each equation and compare the resulting values of y.
For Julie's pool: y = 8,400 + 5.2(30)
= 8,400 + 156
= 8,556 gallons.
For Elaina's pool, we need to calculate the rate of filling at t = 7 minutes to determine the constant rate:
Rate of filling for Elaina's pool from t = 7 to t = 30: 4.8 gallons per minute.
Therefore, Elaina's pool will contain an additional 4.8 gallons per minute for the remaining 23 minutes.
At t = 7 minutes, Elaina's pool contained 7,883.6 gallons.
Additional water added by Elaina's pool from t = 7 to t = 30:
4.8 gallons/minute × 23 minutes = 110.4 gallons.
Total water in Elaina's pool after 30 minutes: 7,883.6 gallons + 110.4 gallons
= 7,994 gallons.
Therefore, after 30 minutes, Julie's pool will contain more water than Elaina's pool.
To learn more on slope intercept form click:
https://brainly.com/question/9682526
#SPJ4
Julie's family is filling up the pool in her backyard. The equation y=8,400+5. 2x can be used to show the rate of which the pool is filling up
Where y is the total amount of water (gallons) and x is the amount of time (minutes). Her neighbor Elaina is also filling up the pool as shown in the table below.
Min 0 3 5 7
GAL 7850 7864.4 7874 7883.6
a) Whose pool is filling at a faster rate?
b)Whose pool initially contained more water?explain.
c) After 30 minutes, whose pool will contain more water?
Alan Will Throw A Six-Sided Fair Die Repeatedly Until He Obtains A 2. Bob Will Throw The Same Die Repeatedly Unit He Obtains A 2 Or 3. We Assume That Successive Throws Are Independent, And Alan And Bob Are Throwing The Die Independently Of One Another. Let X Be The Sum Of Numbers Of Throws Required By Alan And Bob. A) Find P(X=9) B) Find E(X) C) Find Var(X)
Alan will throw a six-sided fair die repeatedly until he obtains a 2. Bob will throw the same die repeatedly unit he obtains a 2 or 3. We assume that successive throws are independent, and Alan and Bob are throwing the die independently of one another. Let X be the sum of numbers of throws required by Alan and Bob.
a) Find P(X=9)
b) Find E(X)
c) Find Var(X)
A. [P(X=9) = \frac{1}{6}\cdot\frac{1}{3} + \frac{2}{4}\cdot\left(\frac{5}{6}\right)^8 \approx 0.012]
B. [E(X) = E(X_A) + E(X_B) = 6+3 = 9]
C. The numbers of throws required by Alan and Bob are independent geometric random variables,
a) To find P(X=9), we need to consider all possible ways that Alan and Bob can obtain a 2 or 3 on their ninth throw, while not obtaining it on any previous throws. Note that Alan and Bob may obtain the desired outcome on different throws.
For example, one possible sequence of throws for Alan is: 1, 4, 5, 6, 6, 1, 2, 3, 6. And one possible sequence of throws for Bob is: 2, 4, 5, 5, 1, 3, 2, 2, 2. In this case, X = 9 because Alan required 9 throws to obtain a 2, and Bob obtained a 2 on his ninth throw.
There are many other possible sequences of throws that could result in X = 9. We can use the multiplication rule of probability to calculate the probability of each sequence occurring, and then add up these probabilities to obtain P(X=9).
Let A denote the event that Alan obtains a 2 on his ninth throw, and let B denote the event that Bob obtains a 2 or 3 on his ninth throw (given that he did not obtain a 2 or 3 on any earlier throw). Then we have:
[P(X=9) = P(A \cap B) + P(B \cap A^c) + P(A \cap B^c)]
where (A^c) denotes the complement of event A, i.e., Alan does not obtain a 2 on his first eight throws, and similarly for (B^c).
Since the die is fair and each throw is independent, we have:
[P(A) = \frac{1}{6},\quad P(A^c) = \left(\frac{5}{6}\right)^8]
[P(B) = \frac{2}{6},\quad P(B^c) = \left(\frac{4}{6}\right)^8]
Therefore, we can calculate:
[P(A \cap B) = P(A)P(B) = \frac{1}{6}\cdot\frac{1}{3}]
[P(B \cap A^c) = P(B|A^c)P(A^c) = \frac{2}{4}\cdot\left(\frac{5}{6}\right)^8]
[P(A \cap B^c) = P(A|B^c)P(B^c) = 0 \quad (\text{since } A \text{ and } B^c \text{ are mutually exclusive})]
Therefore,
[P(X=9) = \frac{1}{6}\cdot\frac{1}{3} + \frac{2}{4}\cdot\left(\frac{5}{6}\right)^8 \approx 0.012]
b) To find E(X), we use the formula for the expected value of a sum of random variables:
[E(X) = E(X_A) + E(X_B)]
where (X_A) and (X_B) are the numbers of throws required by Alan and Bob, respectively.
Since Alan obtains a 2 with probability (\frac{1}{6}) on each throw, the number of throws required by Alan follows a geometric distribution with parameter (p=\frac{1}{6}). Therefore, we have:
[E(X_A) = \frac{1}{p} = 6]
Similarly, since Bob obtains a 2 or 3 with probability (\frac{2}{6}) on each throw, the number of throws required by Bob also follows a geometric distribution with parameter (p=\frac{2}{6}). However, Bob may obtain a 2 or 3 on his first throw, in which case X_B = 1. Therefore, we have:
[E(X_B) = \frac{1}{p} + (1-p)\cdot\frac{1}{p} = \frac{1}{p}(2-p) = 3]
Therefore, we obtain:
[E(X) = E(X_A) + E(X_B) = 6+3 = 9]
c) To find Var(X), we use the formula for the variance of a sum of random variables:
[Var(X) = Var(X_A) + Var(X_B) + 2Cov(X_A,X_B)]
where (Var(X_A)) and (Var(X_B)) are the variances of the numbers of throws required by Alan and Bob, respectively, and Cov(X_A,X_B) is their covariance.
Since the numbers of throws required by Alan and Bob are independent geometric random variables,
Learn more about number from
https://brainly.com/question/27894163
#SPJ11
Sam pushes crates starting from rest across the floor of his classroom of 3 s with a net force as shown above.
For each crate, rank impulses delivered from greatest to least
For each create, rank changes in momentum from greatest to least
For each crate, rank final speeds grom greatest to least
For each crate, rank momenta in 3 s from greatest to least
To rank the given quantities from greatest to least, we need to understand the definitions of each term:
(a) Impulse delivered: Impulse is defined as the change in momentum of an object. It can be calculated by multiplying the force applied to the object by the time interval over which the force is applied.
(b) Change in momentum: Change in momentum is the difference between the final momentum and initial momentum of an object. It can be calculated by subtracting the initial momentum from the final momentum.
(c) Final speed: The final speed of an object is the magnitude of its velocity at the end of a given time period.
(d) Momentum in 3 s: Momentum is the product of an object's mass and velocity.
- Impulse delivered is directly related to the net force acting on the crate. If the net force is higher, the impulse delivered will also be higher.
- Change in momentum is equal to the impulse delivered to the object. So, the ranking of impulse and change in momentum will be the same.
- Final speed depends on the initial conditions of the crate and the net force acting on it. If the net force is significant and acts in the direction of motion, the final speed will be higher.
- Momentum in 3 seconds depends on the initial momentum, net force, and time. Without specific values, it is challenging to determine the exact ranking.
Learn more about Quantity here:
https://brainly.com/question/14581760
#SPJ11
