The annual profits for a company are given
in the following table, where x represents
the number of years since 2006, and y
represents the profit in thousands of
dollars. Write the linear regression
equation that represents this set of data,
rounding all coefficients to the nearest
hundredth. Using this equation, estimate
the calendar year in which the profits
would reach 482 thousand dollars.
Years since 2006 (x)
0
1
2
3
4
5
Profits (y)
(in thousands of dollars)
147
167
209
239
250
267
The linear regression of the data is given as follows:
y = 25.1x + 150.4.
The estimate for the year in which the profit would reach $482,000 is of:
2019.
How to find the equation of linear regression?To find the regression equation, also called line of best fit or least squares regression equation, we need to insert the points (x,y) in the calculator.
The points for this problem are given as follows:
(0, 147), (1, 167), (2, 209), (3, 239), (4, 250), (5, 267).
Hence the equation is of:
y = 25.1x + 150.4.
The estimate for the year in which the profit would reach $482,000 is x years after 2006, when y = 482, hence:
482 = 25.1x + 150.4.
x = (482 - 150.4)/25.1
x = 13.
2006 + 13 = 2019 -> year.
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geometric: you vs serena williams* you are playing serena williams (who has one hand tied behind her back) in tennis. your probability of losing is 98% and thus probability of winning is 2%. you will stop playing after you beat her. what is the probability you win on the 20th game played?
Probability determines the likelihood of an event occurring: P(A) = f / N. Odds and probability are related but odds depend on the probability. You first need probability before determining the odds of an event occurring.
P(A) = f / N.
probability = 2%/100=0.0002
One of the areas of probability theory is the estimation of the chance of experiments occurring. Using a probability, we can calculate everything from the likelihood of getting heads or tails when flipping a coin to the likelihood of making a research error, for example. It is essential to appreciate the most basic definitions of this branch, such as the formula for computing probabilities in equiprobable sample spaces, the likelihood of two events joining together, the probability of the complementary event, etc., in order to properly understand it .Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. Mathematics has incorporated probability to forecast the likelihood of various events.
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A bag contains 2 red marbles and 3 black marbles. If Abby picks a marble without looking, returns it to the bag, and then draws a second marble, what is the probability that both marbles are red? Write your answer as a fraction.
The chance of a series of draws can be calculated while drawing from a collection of things (for instance, a deck of cards) without replacing the items after they are drawn.
What are the parameter for calculating the probability?Calculate the likelihood of the initial draw while accounting for the sample space's abundance of favourable outcomes. The probability of drawing two consecutive red marbles is 4/25.
Probabilities are always in the 0 to 1 range. Probability = Number of favourable outcomes / Total number of outcomes is the general formula for probability. Or. P(A) = f / N.
On the initial draw, there is a 2/5 chance of getting a red marble.
Out of the five marbles, there are two red ones.
On the second draw, there is still a 2/5 chance of drawing a red marble.
Probability of drawing a red marble on the first draw equals the probability of drawing two red marbles (Probability of drawing a red marble on the second draw)
= (2/5) × (2/5)
= 4/25
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If 1 pound of chocolate cream at Philadelphia candies cost $7. 52. How much does that candy cost by the ounce
The cost of one ounce of chocolate cream at Philadelphia according to the cost in pound will be $0.47.
We are aware of the fact that 1 pound is equal to 16 ounces. So, the cost of one pound will be the cost of 16 ounces. Representing this information in ounces and expression -
Cost of 1 pound = 7.52
Rewriting the equation according to ounce
Cost of 16 ounces = 7.52
So, cost of 1 ounce = 7.52/16
Performing division on Right Hand Side of the equation
The cost of one ounce = 0.47
Therefore, the cost of one ounce will be $0.47.
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how many modes are there in the following data set, which represents the number of days worked in october for 10 programmers of a software company? data set: 7,20,22,17,19,15,27,30,14,7
To mode of given data set of number of days worked in october for 10 programmers of a software company is 7 and 14.
The mode of a data set is the value that appears most frequently. In this case, we can see that both 7 and 14 appear twice, while all other values appear only once. Therefore, there are two modes in this data set: 7 and 14.
To solve mathematically, we can use the mode formula:
mode = value with the highest frequency
In this case, we have two values with the highest frequency, which are 7 and 14. Therefore, the mode of given data set is 7 and 14.
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The table below shows the cost of mailing a postcard in different years. During
which time interval did the cost increase at the greatest average rate?
Answer: 1898-1971
Step-by-step explanation: find the difference between the years and divide it over the change in cost
1. difference of yrs. : 1971-1898 = 73
2. difference of cost : 6-1 = 5
3. divide : 73/5 = 14.6 (avg)
avg for #1) 14.6
avg of #2) 1
avg of #3) 2.1
avg of #4) approx. 0.55
Jonah has $7,586 in an account that earns 10% interest compounded annually.
In a case whereby Jonah has $7,586 in an account that earns 10% interest compounded annually, the amount he will have in 3 years is $10,096.97
How can the amount he will have in 3 years be calculated?The concept that will be used here is counpound interest.
The amount that will be present in the account in 3 years can be estimated using the expression beleow which is:
FV = P(1 + r)^n
P represent the amount in the account which is $7,586
r represen the interest rate which is 10/100 =0.01
n represent the number of years
FV represent the future value
We can input all these values as :
FV =[ 7,586(1 + 0.1)^3]
= (7,586 x 1.1^3 )
= $10,096.97
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complete question:
Jonah has $7,586 in an account that earns 10% interest compounded annually.
To the nearest cent, how much will he have in 3 years?
Sam bought six pens for an unknown price plus a backpack. If he spent a total of $28.50, write an equation to find the cost of each pen.
Answer: $4.75 for each pen
Step-by-step explanation:
$28.50/6= 4.75$
10 The number of students who brought
their own lunch to school on Tuesday was"
57 fewer than the number of students
who ate the school lunch. There were
more than 485 students who brought
their lunch.
Part A
Write an inequality to find the possible
number of students, n, who ate the
school lunch.
Part B
How many students could have possibly
eaten the school lunch?
Part A: The inequality to find the possible number of students, n, who ate the school lunch n > 542 students
Part B: The number of students, that possibly eat lunch at school is 542 or more students
What is inequality?Inequality is defined as the relation which makes a non-equal comparison between two given functions.
Part A; The number of students who bring lunch, B > 485 students
The number of students who eat lunch in school, E = B + 57
∴ E > 485 + 57
The number of students who can possibly eat lunch in school;
E > 542 students
Part B;
The number of students, n, that possibly eat lunch at school, are;
n = 542 or more students.
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what is the minimum value of y=x^2+36/5
Answer:
(36/5)
Step-by-step explanation:
y=x^2+36/5
Since x is squared, the result will always be a positive number, regardless of the actual value of x. Starting with a value of x of zero, the x^2 term will only grow as one moves away from 0. E.g., (-1)^2 = 1, (-2)^2 = 4, etc.
So the minium for this equation will occur when x = 0. It only gets larger from that point. The minimum is (36/5).
See the attached graph.
Need help looking for the answer!!
Answer: it will take both the kids 5 hours
Step-by-step explanation: 45+5x5=50 5+9x5=5
9 times 5 is 45 in 5 hours Isaiah will have completed 5 pages and he started at 45 so in 5 hours Brenna will have completed 45 pages and she started at 5
Please help thank you
Raoul and Sven are at an angle of = cos1 (cos a cos 43° + sin a sin 43°). The sled is being moved by a force with a magnitude of [(105 cos a + 122 cos 43°)2 + (105 sin a + 122 sin 43°)2].
What happens when two vectors are combined?The outcome of two vectors is determined by the parallelogram law of vector addition. The resultant vector, R=(A2+B2+2ABcos), follows this law. A and B are the two vectors in this situation, and is the angle between them.
The component in the direction of the sled's path is:
105 cos a
The component perpendicular to the path is:
105 sin a
Similar to this, Sven's force may be broken down into two parts: one that is parallel to the sled's course and one that is perpendicular to it. The element pointing in the sled's direction is:
122 cos 43°
The component perpendicular to the path is:
122 sin 43°
The net force on the sled is the vector sum of these four components:
Net force = (105 cos a + 122 cos 43°) i + (105 sin a + 122 sin 43°) j
The magnitude of the net force is:
|Net force| = √[(105 cos a + 122 cos 43°)² + (105 sin a + 122 sin 43°)²]
To find the angle between Raoul and Sven, we can use the dot product of their forces:
Raoul · Sven = |Raoul| |Sven| cos θ
We can solve for θ as follows:
cos θ = (Raoul · Sven) / (|Raoul| |Sven|)
cos θ = [(105)(122) cos a cos 43° + (105)(122) sin a sin 43°] / (105)(122)
cos θ = cos a cos 43° + sin a sin 43°
θ = cos⁻¹(cos a cos 43° + sin a sin 43°)
a. The angle between Raoul and Sven is θ = cos⁻¹(cos a cos 43° + sin a sin 43°).
b. The magnitude of the force moving the sled is |Net force| = √[(105 cos a + 122 cos 43°)² + (105 sin a + 122 sin 43°)²].
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Michael took his three kids to get haircuts the haircuts cost $57. 50 he also gave the stylist a 20% tip what was the total cost including tip for Michael’s three kids to get haircuts
The total cost of haircuts for Michael's three children, including tip, would be $207.00.
If Michael gave a 20% tip and the haircuts cost $57.50, the tip would be:
Tip = 0.20 * $57.50 = $11.50
The total cost of the haircuts, including the tip, is as follows:
Total cost = haircut cost + tip
The total cost is $57.50 + $11.50.
The total cost is $69.00.
Since Michael took his three children to get haircuts, the total cost for all three children, including tip, would be:
Total cost for three children = three times the total cost
Total cost for three children = three times $69.00
The total cost for three children is $207.00.
As a result, the total cost of haircuts for Michael's three children, including tip, would be $207.00.
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Which pairs of rectangles are similar polygons?
Select each correct answer.
Answer:
The corrects answers are B and D
Since you divide the same side for both the the rectangles on both parts and if the division statements are equaled, then they are similar.
Step-by-step explanation:
have a nice day.
What information does the source supply?
The information provided by a source in a graph is C. Who put the data together.
How does a source contribute to a graph ?The information provided by a source in a graph can include various aspects such as the data being displayed, the method used to collect the data, and who put the data together (also known as the data source).
The data source is important because it provides context and credibility to the data being presented. It helps to understand the reliability and accuracy of the data, as well as any potential biases that may be present. In this case, the source is the Human Resources Department.
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Translate the following sentence into an equation. Then, SOLVE the equation: the difference between c and 25 is 75
To translate the sentence into an equation, we can let c represent the value we want to find. The difference between c and 25 is 75, so we can write this as:
c - 25 = 75
To solve the equation, we can add 25 to both sides:
c - 25 + 25 = 75 + 25
c = 100
So the value of c is 100.
a basketball player made 17 out of 20 free throws the practice. what percent of the free throws did the player miss?
Answer:
85%
Step-by-step explanation:
well, she missed 3 throws only from all 20's, hmmm if we take 20(origin amount) to be the 100%, what's 3 off of it in percentage?
[tex]\begin{array}{ccll} amount&\%\\ \cline{1-2} 20 & 100\\ 3& x \end{array} \implies \cfrac{20}{3}~~=~~\cfrac{100}{x} \\\\\\ 20x=300\implies x=\cfrac{300}{20}\implies x=15[/tex]
Microsoft Windows XP operating system is installed on 20% of the company's computers, Microsoft Windows 7 is installed on 85% of computers, and Linux operating system is installed on 10%. At the same time, Linux and Microsoft Windows 7 are installed on 6% of computers, Microsoft Windows XP and Linux on 4%, all three programs are installed on 2% of computers. How many percent of computers have the Microsoft operating system installed?
To find out how many percent of computers have the Microsoft operating system installed, we need to add up the percentage of computers that have Windows XP and Windows 7 installed.
First, let's find out how many percent of computers have both Windows 7 and Linux installed:
85% (Windows 7) - 6% (Windows 7 & Linux) = 79% (Windows 7 only)
Next, let's find out how many percent of computers have both Windows XP and Linux installed:
20% (Windows XP) - 4% (Windows XP & Linux) = 16% (Windows XP only)
Finally, let's add up the percentage of computers that have Windows XP and Windows 7 installed:
79% (Windows 7 only) + 16% (Windows XP only) = 95%
So, 95% of the company's computers have the Microsoft operating system installed.
Answer:
o calculate the percentage of computers with the Microsoft operating system installed, we need to find the total number of computers that have either Windows XP or Windows 7 installed.
Microsoft Windows XP operating system is installed on 20% of the company's computers, so 20% of the computers have Windows XP installed.
Microsoft Windows 7 is installed on 85% of computers, so 85% of the computers have Windows 7 installed.
Linux and Microsoft Windows 7 are installed on 6% of computers, so 6% of the computers have both Windows 7 and Linux installed.
Microsoft Windows XP and Linux on 4% of computers, so 4% of the computers have both Windows XP and Linux installed.
All three programs are installed on 2% of computers, so 2% of the computers have all three programs installed.
To avoid double-counting the computers that have both Windows 7 and Linux installed, we need to subtract 6% from the total number of computers with Windows 7 installed. The same applies to the computers that have both Windows XP and Linux installed, so we need to subtract 4% from the total number of computers with Windows XP installed.
The total number of computers with Windows XP or Windows 7 installed is 20% + (85% - 6%) + (20% - 4%) = 20% + 79% + 16% = 115%.
So, 115% of the company's computers have the Microsoft operating system installed.
Step-by-step explanation:
A culture of bacteria has an initial population of 8300 bacteria and
doubles every 10 hours. Using the formula Pt = P0 2^t/d, where Pt is
the population after t hours, P0 is the initial population, t is the time in
hours and d is the doubling time, what is the population of bacteria in the culture after 3 hours, to the nearest whole number?
Bryanna's younger sister is stacking her toy blocks on top of each other to build a tall tower. If each block is 2.4 inches long, how many blocks will she have stacked if the tower reaches a height of 4 fe
Use Table B in your Student Guide to answer the questions about ion concentrations.
A solution with a pH = 13 has approximately how many moles of OH ions per liter?
How many moles of H* would this same solution have per liter?
(Use the decimal form of your answer.)
A different solution with an H+ concentration of 1.0 x 10-4 would have a pH =
Considering the definition of pH and pOH, you obtain that:
a solution with a pH = 13 has 0.1 moles of OH⁻ ions per liter.a solution with a pH = 13 has 1×10⁻¹³ moles of H⁺ ions per liter.a different solution with an H⁺ concentration of 1.0×10⁻⁴ would have a pH= 4.What is the pH?pH is the Hydrogen Potential. It is a measure of acidity or alkalinity that indicates the amount of hydrogen ions present in a solution or substance.
Mathematically, pH is calculated as the negative base 10 logarithm of the activity of hydrogen ions, that is, the concentration of hydrogen ions:
pH= - log [H⁺]
Similarly, pOH is a measure of hydroxyl ions in a solution and is expressed as the logarithm of the concentration of OH⁻ ions, with the sign changed:
pOH= - log [OH⁻]
The following relationship can be established between pH and pOH:
pOH + pH= 14
In this case, you know that a solution has a pH = 13. So, the pOH can be calculated as:
pOH + 13= 14
pOH= 14 -13
pOH= 1
Then, the concentration of OH⁻ can be calculated as:
1= - log [OH⁻]
[OH⁻]= 10⁻¹
[OH⁻]= 0.1
Finally, a solution with a pH = 13 has 0.1 moles of OH⁻ ions per liter.
On the other side, remembering that pH = 13, the concentration of H⁺ can be calculated as:
13= - log [H⁺]
[H⁺]= 10⁻¹³
[H⁺]= 1×10⁻¹³
So, a solution with a pH = 13 has 1×10⁻¹³ moles of H⁺ ions per liter.
Finally, you have a different solution with an H⁺ concentration of 1.0×10⁻⁴. Replacing in the definition of pH:
pH= - log (1.0×10⁻⁴)
Solving:
pH= 4
Then, a different solution with an H⁺ concentration of 1.0×10⁻⁴ would have a pH= 4.
In summary, you obtain that:
a solution with a pH = 13 has 0.1 moles of OH⁻ ions per liter.a solution with a pH = 13 has 1×10⁻¹³ moles of H⁺ ions per liter.a different solution with an H⁺ concentration of 1.0×10⁻⁴ would have a pH= 4.Read more about molar mass here:
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1 The total cost of 5 shirts and 3 pairs of pants is $206. The total cost of a
shirt and a pair of pants is $48. How much does a pair of pants cost?
The cost of a pair of pants is $17.
And the cost of a shirt is $31.
What is the system of equations?One or many equations having the same number of unknowns that can be solved simultaneously are called simultaneous equations. And the simultaneous equation is the system of equations.
Given:
The total cost of 5 shirts and 3 pairs of pants is $206.
The total cost of a shirt and a pair of pants is $48.
Let y be the cost of a pair of pants and x be the cost of a shirt.
That means,
we have the system of equations,
x + y = 48 {equation 1}
5x + 3y = 206 {equation 2}
Multiply 5 to the equation 1 and then subtract equation 2 to equation 1,
we get,
2y = 34
y = 17
And x = 48 - 17 = 31.
Hence, $17 is the cost of a pair of pants.
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find the values of the variables for the parallelogram
The value of m is 40 degrees, the value of x is 6 and value of y is 9.
What is Parallelogram?Parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.
To find the value of variable m.
Forming an equation:
3m = 120°
m = 120/3
m = 40
Therefore the value of m is 40.
Finding x:
x is opposite to the side of measure 6.
As per the first property.
x = 6
So, x = 6
Therefore the value of x is 6.
Finding y.
5y is opposite the angle measuring 45°.
Forming an equation.
5y = 45°
y = 45/5
y = 9
Therefore the value of y is 9.
Hence, the value of m is 40 degrees, the value of x is 6 and value of y is 9.
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The complete question is given below and in attachment
Find the value of each variable in the parallelogram. explain step by step
Please help! I don't understand how my teacher does the work, so therefore, I don't understand how to do these word problems. 100-point jackpot for answering.
Answer: 1) 4 $3 and 4 $2; 2) 4 candles; 3) 4 adults and 6 students; 4) Small is 11.67 and Large is 83.75
Step-by-step explanation: 1. Total of $20 and needs 9 donuts. One is $3 and one is $2. Start with 3 $3 that 9 and let's do 3 for $2 which is 6. 9+6 is $15. 20-15 is 5. We have $5 left so we can buy in total 4 $3 donuts and 4 $2 donuts.
2. She has to make at least 4 candles. Booth is $40. And $2 per candle. 4*2 is $8. 40+8 = 48 and 48/12 is 4 candles.
3. They have a total of 120 sold. So let's start with adult tickets: $15 each. So let's start with 4 sold. 15*4 = 60. Students: $10 each. 10*4 is $40. 60+40 is 100 sold! So let's do 2 more student tickets to get 120 sold. 4 adults sold and 6 students sold.
4. Let's work with 6 and 8 you set up the equation 6x+8y=670 you get x = 11.67 and y is 83.75. So the large price is $83.75 and the Small is $11.67
10+3s≥19 Need help pls
Answer: s≥3
Step-by-step explanation: fist you need to get the variable by itself on one side. So subtract the 10 from each side and you have 3s≥9. Then you divide each side ny 3 to get s by itself, divide 9 by 3. Then your answer is s≥3
Find the 92nd term of the arithmetic sequence 8, 25, 42, ...8,25,42,
The 92nd term of the arithmetic sequence 8, 25, 42, ..... is equal to 1,555.
How to calculate an arithmetic sequence?Mathematically, the nth term of an arithmetic sequence can be calculated by using this expression:
aₙ = a₁ + (n - 1)d
Where:
d represents the common difference.a₁ represents the first term of an arithmetic sequence.n represents the total number of terms.Next, we would determine the common difference as follows.
Common difference, d = a₂ - a₁
Common difference, d = 25 - 8
Common difference, d = 17
Substituting the given parameters into the formula, the 92nd term of this arithmetic sequence is given by;
a₉₂ = a₁ + (92 - 1)d
a₉₂ = 8 + (91)(17)
a₉₂ = 8 + 1,547
a₉₂ = 1,555.
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An aeroplane flies 1400km in 1h 45mins.how far will it fly in 2h 30min at the same avarage speed?
Answer:
The aeroplane will fly 2000 km if travelling at the same average speed.
Step-by-step explanation:
To calculate the distance an aeroplane will fly in 2 h 30 min if it flies 1400 km in 1 h 45 mins, use the Speed Distance Time formula.
[tex]\boxed{\sf Speed=\dfrac{Distance}{Time}}[/tex]
There are 60 minutes in one hour. Therefore:
1 h 45 mins = 1.75 hours2 h 30 mins = 2.5 hoursTo calculate the speed of the aeroplane, divide the distance by the time it takes to travel that distance. Therefore, if an aeroplane flies 1400 km in 1.75 hours then its average speed is:
[tex]\sf Speed=\dfrac{1400\;km}{1.75\;h}=800\;km/h[/tex]
To determine the distance the aeroplane will travel in 2.5 hours, multiply the found speed of 800 km/h by the time:
[tex]\sf Distance=800\;km/h \times 2.5\;h=2000\;km[/tex]
Therefore, the aeroplane will fly 2000 km in 2 h 30 min if travelling at the same average speed.
A set of eight cards were labeled with D, I, V, I, S, I, O, N. What is the sample space for choosing one card?
S = {V, S, I, D, I, I, N, O}
S = {D, I, S, O, N}
S = {V, I, O}
S = {D, I}
The sample space for choosing one card is S={V, S, I, D, I, I, N, O}.
What is a sample space?
A set of potential outcomes from a random experiment is known as a sample space. The sample space is identified by the letter "S". Events are a subset of the potential outcomes of an experiment. The results in a sample area could vary depending on the experiment.
Given that, a set of eight cards were labeled D, I, V, I, S, I, O, and N.
All the possible outcomes should be included in the sample space.
Here, the sample space will be { D, I, V, I,S, I, O, N}
Elements in the order will be
S={V, S, I, D, I, I, N, O}
Therefore, the sample space for the given set of cards is
S={V, S, I, D, I, I, N, O}.
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HELP ME!
A passenger train leaves a train depot 2 h after a freight train leaves the same depot. The freight train is traveling 28 mph slower than the passenger train. Find the rate of each train if the passenger train overtakes the freight train in 3 h.
A. Passenger train ___ mph
B. freight train ___ mph
Answer: A. Passenger train: 52 mph B. Freight train: 24 mph
Step-by-step explanation:
Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is the following. F(x) = 0 x < 0 x2 25 0 ≤ x < 5 1 5 ≤ x Use the cdf to obtain the following. (If necessary, round your answer to four decimal places.)
(a) Calculate P(X ≤ 3).
(b) Calculate P(2.5 ≤ X ≤ 3).
(c) Calculate P(X > 3.5).
(d) What is the median checkout duration ? [solve 0.5 = F()].
(e) Obtain the density function f(x). f(x) = F '(x) =
(f) Calculate E(X).
(g) Calculate V(X) and σx. V(X) = σx =
(h) If the borrower is charged an amount h(X) = X2 when checkout duration is X, compute the expected charge E[h(X)].
By Using the Cumulative distributed Function(CDF)m we get:
a) P (x <= 3 ) = 0.36
b) P ( 2.5 <= x <= 3 ) = 0.11
c) P (x > 3.5 ) = 1 - 0.49 = 0.51
d) x = 3.5355
e) f(x) = x / 12.5
f) E(X) = 3.3333
g) Var (X) = 13.8891 , s.d (X) = 3.7268
h) E[h(X)] = 2500
Given:
The cdf is as follows:
F(x) = 0 x < 0
F(x) = (x^2 / 25) 0 < x < 5
F(x) = 1 x > 5
Now,
a) Evaluate the CDF given with the limits 0 < x < 3.
So, P (x <= 3 ) = (x^2 / 25) | 0 to 3
P (x <= 3 ) = (3^2 / 25) - 0
P (x <= 3 ) = 0.36
b) Evaluate the CDF given with the limits 2.5 < x < 3.
So, P ( 2.5 <= x <= 3 ) = (x^2 / 25) | 2.5 to 3
P ( 2.5 <= x <= 3 ) = (3^2 / 25) - (2.5^2 / 25)
P ( 2.5 <= x <= 3 ) = 0.36 - 0.25 = 0.11
c) Evaluate the CDF given with the limits x > 3.5
So, P (x > 3.5 ) = 1 - P (x <= 3.5 )
P (x > 3.5 ) = 1 - (3.5^2 / 25) - 0
P (x > 3.5 ) = 1 - 0.49 = 0.51
d) The median checkout for the duration that is 50% of the probability:
So, P( x < a ) = 0.5
⇒ (x² / 25) = 0.5
⇒ x² = 12.5
⇒ x = 3.5355
e) The probability density function can be evaluated by taking the derivative of the cdf as follows:
pdf f(x) = d(F(x)) / dx = x / 12.5
f) The expected value of X can be evaluated by the following formula from limits - ∞ to +∞:
E(X) = integral ( x . f(x)).dx limits: - ∞ to +∞
E(X) = integral ( x² / 12.5)
E(X) = x³ / 37.5 limits: 0 to 5
E(X) = 5³ / 37.5 = 3.3333
g) The variance of X can be evaluated by the following formula from limits - ∞ to +∞:
Var(X) = integral ( x² . f(x)).dx - (E(X))^2 limits: - ∞ to +∞
Var(X) = integral ( x³ / 12.5).dx - (E(X))^2
Var(X) = x⁴/ 50 | - (3.3333)^2 limits: 0 to 5
Var(X) = 5⁴/ 50 - (3.3333)^2 = 13.8891
Standard Deviation (s.d) (X) = sqrt (Var(X)) = sqrt (13.8891) = 3.7268
h) Find the expected charge E[h(X)] , where h(X) is given by:
h(x) = (f(x))^2 = x^2 / 156.25
The expected value of h(X) can be evaluated by the following formula from limits - ∞ to +∞:
E(h(X))) = integral ( x . h(x) ).dx limits: - ∞ to +∞
E(h(X))) = integral ( x^3 / 156.25)
E(h(X))) = x^4 / 156.25 limits: 0 to 25
E(h(X))) = 25^4 / 156.25 = 2500
Learn more about cumulative distribution function:
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