We are given a random variable X representing the length of time (in years) that a laptop lasts. The probability density function (PDF) of X is given by f(x) = 1/(6√x).
The probability density function (PDF) describes the likelihood of different values of a continuous random variable. In this case, the PDF of X is given as f(x) = 1/(6√x).
To understand the PDF, we need to consider its properties. First, note that the PDF is only defined for x ≥ 0, as the square root term requires non-negative values. Second, the PDF is always positive for valid values of x, indicating that the probability density is non-zero for all possible values of X.
The function f(x) = 1/(6√x) is a decreasing function of x. As x increases, the denominator √x also increases, leading to smaller values of f(x). This implies that the probability density decreases as the length of time (x) increases, which is reasonable since it is less likely for a laptop to last for a longer period.
The PDF can be used to calculate probabilities and expected values associated with the random variable X. For example, to find the probability that the laptop lasts between a and b years, we can integrate the PDF over the interval [a, b]. The expected value of X, denoted as E(X), can be calculated by integrating x·f(x) over all possible values of x.
Overall, the PDF f(x) = 1/(6√x) provides a mathematical description of the probability distribution of the length of time that a laptop lasts, capturing the decreasing likelihood as the lifespan increases.
Learn more about probabilities here:
https://brainly.com/question/32117953
#SPJ11
1. What type of function is B(x), linear, quadratic or exponential? Justify your answer and
show calculations to support your conclusion.
Years (x)
0
1
2
3
45
5
ANSWER:
Batana
B(x)
2
6
18
54
162
486
RATIOS
FIRST
DIFFERENCES
SECOND
DIFFERENCES
The function B(x) is an exponential function.
How to determine the type of functionTo determine the type of function B(x), we can examine the given values of x and the corresponding values of B(x).
Years (x) B(x)
0 2
1 6
2 18
3 54
4 162
5 486
By analyzing the values, we can observe that the ratio of B(x) to B(x-1) is constant, which indicates an exponential relationship. Let's calculate the ratios:
B(1)/B(0) = 6/2 = 3
B(2)/B(1) = 18/6 = 3
B(3)/B(2) = 54/18 = 3
B(4)/B(3) = 162/54 = 3
B(5)/B(4) = 486/162 = 3
The ratios are all equal to 3, which suggests exponential growth. Therefore, B(x) is an exponential function.
Learn more about exponential function at
https://brainly.com/question/2456547
#SPJ1
Find the limit of the sequence { 2
, 2 2
, 2 2 2
,…}
The limit of the sequence {2, 2², 2³, ...} does not exist. The sequence diverges.
The given sequence is {2, 2², 2³, ...}, where each term is the result of raising 2 to an increasing power. To find the limit of this sequence, we need to determine the behavior of the terms as the index increases without bound.
As the index n increases, the terms of the sequence become larger because each term is obtained by multiplying the previous term by 2. This means that each term is double the previous term.
Since the base value 2 is positive and greater than 1, raising 2 to increasingly larger powers will result in the terms of the sequence growing without bound. In other words, the terms become arbitrarily large as the index n increases.
In summary, the terms of the sequence increase exponentially as the index increases, and there is no finite value that the terms converge to. The sequence grows without bound, indicating that the limit is undefined.
The complete question is:
Find the limit of the sequence {2, 2², 2³, ...}
To know more about sequence:
https://brainly.com/question/30262438
#SPJ4
A particle is moving with the given data. Find the position of the particle. a(t) = t²9t + 8, s(0) = 0, s(1) = 20 s(t) X Need Help?
Given that a(t) = t²9t + 8, s(0) = 0, s(1) = 20. To find the position of the particle, we need to integrate the given acceleration with respect to time to get the velocity function and integrate the velocity function to get the position function. Therefore, the position of the particle is 22.4 when t = 1.
Step 1: Finding the Velocity Function v(t) = ∫a(t) dt= ∫(t²9t + 8) dt= [t³/3 * 9/3t² + 8t] + C1= 3t⁴/4 + 8t + C1where C1 is the constant of integration.
To find C1, we use the initial condition s(0) = 0v(0) = 3(0)⁴/4 + 8(0) + C1C1 = 0
Hence the velocity function isv (t) = 3t⁴/4 + 8t.
Step 2: Finding the Position Function.
s(t) = ∫v(t) dt= ∫(3t⁴/4 + 8t) dt= 3/5 t⁵/4 + 4t² + C2 where C2 is the constant of integration.
To find C2, we use the initial condition s(1) = 20s(1) = 3/5 (1)⁵/4 + 4(1)² + C2C2 = 20 - 3/5 - 4
Hence the position function iss (t) = 3/5 t⁵/4 + 4t² + 92
Putting t = 1, we get the position of the particle.
s(1) = 3/5 (1)⁵/4 + 4(1)² + 92= 3/5 + 4 + 92= 112/5 = 22.4
Therefore, the position of the particle is 22.4 when t = 1.
Therefore, the position of the particle is 22.4 when t = 1.
To know more about Function visit :
https://brainly.com/question/30721594
#SPJ11
h(x)=∑ k=1
[infinity]
b k
(t)sin( 4
πkx
) Please choose the correct option: does your answer only include odd values of k, even values of k, or all values of k ? b k
(t)= Which values of k should be included in this summation? A. Only the even values B. Only the odd values C. All values
All values of k should be included in this summation.
The given function is
[tex]h(x)=∑ k=1[infinity]bk(t)sin(4πkx) Where bk(t)[/tex]
represents the coefficient of the kth term.
The given function can be written as:
[tex]h(x)=b1(t)sin(4πx)+b2(t)sin(8πx)+b3(t)sin(12πx)+.....and so on.[/tex]
Thus, the function h(x) includes all values of k since the sum is being taken over all values of k (from 1 to infinity).
Therefore, the correct answer is option C:
All values of k should be included in this summation.
the given problem is as follows:
Function h(x) is given by
[tex]h(x)=∑ k=1[infinity]bk(t)sin(4πkx).[/tex]
This function can be rewritten as
[tex]h(x)=b1(t)sin(4πx)+b2(t)sin(8πx)+b3(t)sin(12πx)+.....and so on.[/tex]
Here, bk(t) represents the coefficient of the kth term.
Since the summation is being taken over all values of k (from 1 to infinity), it is clear that the function h(x) includes all values of k.
Thus, the correct answer is option C: All values of k should be included in this summation.
Learn more about function
brainly.com/question/31062578
#SPJ11
proove the identity.
(csc()+cot()) (csc(0)-cot(0))=1
To prove the identity `(csc() + cot()) (csc(0) - cot(0)) = 1`, we need to make use of the trigonometric identities. Below is the complete explanation for the given identity and how to prove it:Identity: `(csc() + cot()) (csc(0) - cot(0)) = 1`
We know that the cosecant is the reciprocal of the sine and the cotangent is the reciprocal of the tangent, so we can rewrite the identity as follows:`((1/sin() + cos()/sin()) * (1/sin(0) - cos(0)/sin(0)) = 1
`We need to simplify the expression using the following identities:For the product of two fractions, we multiply the numerators together and multiply the denominators together.`a/b * c/d = (a * c)/(b * d)`The reciprocal of a fraction is flipping the numerator and denominator.`1/x = x^(-1)
`The Pythagorean identity relates the trigonometric functions sine and cosine to the unit circle and is defined as:`sin^2(x) + cos^2(x) = 1`Applying these identities, we can simplify the expression as follows:`(1 + cos() * sin()) / sin() * (1 - cos(0) * sin(0)) / sin(0) = 1`Using the Pythagorean identity, we know that `sin()^2 + cos()^2 = 1` and `sin(0)^2 + cos(0)^2 = 1`, which means that `cos() * sin()` and `cos(0) * sin(0)` are both equal to `sqrt(1 - sin()^2)` and `sqrt(1 - sin(0)^2)`, respectively.
Substituting these values, we get:`(1 + sqrt(1 - sin()^2)) / sin() * (1 - sqrt(1 - sin(0)^2)) / sin(0) = 1`Simplifying this expression, we get:`(sin()^2 + sin() * sqrt(1 - sin()^2)) / sin() * (sin(0)^2 - sin(0) * sqrt(1 - sin(0)^2)) / sin(0) = 1`We can simplify further by canceling out the terms in the numerator and denominator:`sin() + sqrt(1 - sin()^2) = sin(0) + sqrt(1 - sin(0)^2)`Thus, we have proved the identity.
Therefore, the given identity `(csc() + cot()) (csc(0) - cot(0)) = 1` has been proved as `(sin() + sqrt(1 - sin()^2)) / sin() * (sin(0)^2 - sin(0) * sqrt(1 - sin(0)^2)) / sin(0) = 1`.
To know more about identity visit
https://brainly.com/question/27666191
#SPJ11
Debt payments of $610 due today, $1,725 due in 61 days, and
$1,270 due in 350 days respectively are to be combined into a
s
300 days from now. What is the single equivalent payment if
the money is val
The correct answer is the single equivalent payment is $3,605.
To find the single equivalent payment for the combined debts, we need to calculate the present value of each individual debt and then sum them up.
Let's denote:
P1 = $610 (due today)
P2 = $1,725 (due in 61 days)
P3 = $1,270 (due in 350 days)
S = Single equivalent payment (due 300 days from now)
We'll use the concept of present value to calculate the equivalent amounts. The present value of a future payment is given by the formula:
[tex]PV = FV / (1 + r)^n[/tex]
where PV is the present value, FV is the future value (amount due), r is the interest rate, and n is the number of periods.
Given that we don't have an interest rate mentioned in the problem, we'll assume no interest for simplicity. Therefore, r = 0.
Now let's calculate the present value of each debt:
PV1 = $610 / [tex](1 + 0)^0[/tex] = $610 (no time has passed)
PV2 = $1,725 / [tex](1 + 0)^61[/tex] = $1,725
PV3 = $1,270 / [tex](1 + 0)^350[/tex] = $1,270
To find the single equivalent payment, we sum up the present values:
S = PV1 + PV2 + PV3 = $610 + $1,725 + $1,270 = $3,605
Therefore, the single equivalent payment is $3,605.
Learn more about compounded here:
https://brainly.com/question/24274034
#SPJ11
Use a calculator to find the function value to four decimal places. \[ \csc 22^{\circ} 21^{\prime} 49^{\prime \prime} \] \( 1.0813 \) \( 2.0813 \) \( 2.6282 \) \( 2.4306 \)
The function value of \(\csc 22^{\circ} 21^{\prime} 49^{\prime \prime}\) is approximately 2.0813.
The cosecant function, denoted as \(\csc\), is the reciprocal of the sine function. To calculate the value of \(\csc 22^{\circ} 21^{\prime} 49^{\prime \prime}\), we first convert the angle from degrees, minutes, and seconds to decimal degrees.
\(22^{\circ} 21^{\prime} 49^{\prime \prime}\) is equivalent to 22.3636 degrees (rounded to four decimal places).
Next, we evaluate the reciprocal of the sine of 22.3636 degrees using a calculator. The result is approximately 2.0813.
Therefore, the function value of \(\csc 22^{\circ} 21^{\prime} 49^{\prime \prime}\) is approximately 2.0813.
to learn more about decimal places click here:
brainly.com/question/29089858
#spj11
of the volunteers donating blood in a clinic, 20% have the rhesus (rh) factor present in their blood. (a) if five volunteers are randomly selected, what is the probability that at least one does not have the rh factor? (round your answer to four decimal places.) (b) if five volunteers are randomly selected, what is the probability that at most four have the rh factor? (round your answer to four decimal places.) (c) what is the smallest number of volunteers who must be selected if we want to be at least 90% certain that we obtain at least five donors with the rh factor? (round your answer up to the nearest donor.) donors you may need to use the appropriate appendix table or technology to answer this question.
The probability of obtaining at least five donors with the Rh factor when 13 volunteers are selected is 0.874.five donors with the Rh factor is 14.
(a) The probability that at least one of five randomly selected volunteers does not have the Rh factor is 0.9996. (b) The probability that at most four of five randomly selected volunteers have the Rh factor is 0.8106.
(c) The smallest number of volunteers who must be selected to be at least 90% certain that we obtain at least five donors with the Rh factor is 14
(a) There is a 80% chance that any given volunteer does not have the Rh factor. So, the probability that all five volunteers do not have the Rh factor is (0.8)^5 = 0.32768. The probability that at least one of the five volunteers does not have the Rh factor is 1 - 0.32768 = 0.9996.
(b) There is a 0.2^5 = 0.032 probability that all five volunteers have the Rh factor. So, the probability that at most four of the five volunteers have the Rh factor is 1 - 0.032 = 0.8106.
(c) We want the probability of obtaining at least five donors with the Rh factor to be at least 0.9. Using the binomial distribution, we can calculate that the probability of obtaining at least five donors with the Rh factor when 13 volunteers are selected is 0.874.
The probability of obtaining at least five donors with the Rh factor when 14 volunteers are selected is 0.941. So, the smallest number of volunteers who must be selected to be at least 90% certain that we obtain at least five donors with the Rh factor is 14.
To know more about probability click here
brainly.com/question/15124899
#SPJ11
Find the radius and interval of convergence for the power series \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(x+5)^{n}}{n !} \)
a) The radius of convergence is 1/3, and the interval of convergence is (-4/3, -2/3). b) The radius of convergence is 3/2, and the interval of convergence is (2, 8). c) The radius of convergence is 1/2, and the interval of convergence is (1/2, 3/2). d) The radius of convergence is infinity, and the interval of convergence is (-∞, +∞).
To find the radius of convergence and interval of convergence of the given power series, we can use the ratio test. The ratio test states that if we have a power series ∑(a_n * x^n), then the series converges if the limit of the absolute value of the ratio of consecutive terms, lim┬(n→∞)|a_(n+1)/a_n|, is less than 1.
a) For the series ∑[tex](3^n / (n^2) * (x+1)^n)[/tex], applying the ratio test:
lim┬(n→∞)|[tex](3^(n+1) / ((n+1)^2) * (x+1)^(n+1)) / (3^n / (n^2) * (x+1)^n)|[/tex]
= lim┬(n→∞)|[tex]3 * ((n^2) / ((n+1)^2)) * (x+1)|[/tex]
= 3 * |x+1| * lim┬(n→∞)|[tex](n^2) / ((n+1)^2)|[/tex]
Taking the limit as n approaches infinity, we have:
lim┬(n→∞)[tex]|(n^2) / ((n+1)^2)| = 1[/tex]
Therefore, the ratio simplifies to 3 * |x+1|. For convergence, we need 3 * |x+1| < 1. So the radius of convergence is 1/3, and the interval of convergence is (-4/3, -2/3).
b) For the series ∑[tex](((2n-1)! / (3^n)) * (x-5)^n),[/tex] applying the ratio test:
lim┬(n→∞)[tex]|((2(n+1)-1)! / (3^(n+1))) * (x-5)^(n+1)) / ((2n-1)! / (3^n)) * (x-5)^n)|[/tex]
= lim┬(n→∞)[tex]|((2n+1)(2n)(2n-1)! / (3 * (n+1)) * (x-5)|[/tex]
= |2x-10| * lim┬(n→∞)|2n+1| / (3 * (n+1))
Taking the limit as n approaches infinity, we have:
lim┬(n→∞)|2n+1| / (3 * (n+1)) = 2/3
Therefore, the ratio simplifies to (2/3) * |2x-10|. For convergence, we need (2/3) * |2x-10| < 1. So the radius of convergence is 3/2, and the interval of convergence is (2, 8).
c) For the series ∑[tex](n * (-2)^n * (x-1)^n)[/tex], applying the ratio test:
lim┬(n→∞)[tex]|((n+1) * (-2)^(n+1) * (x-1)^(n+1)) / (n * (-2)^n * (x-1)^n)|[/tex]
= lim┬(n→∞)|-2 * (x-1)|
= 2|x-1|
For convergence, we need 2|x-1| < 1. So the radius of convergence is 1/2, and the interval of convergence is (1/2, 3/2).
d) For the series ∑[tex](e^n / (n! * (x+2)^n))[/tex], applying the ratio test:
lim┬(n→∞)[tex]|(e^(n+1) / ((n+1)! * (x+2)^(n+1))) / (e^n / (n! * (x+2)^n))|[/tex]
= lim┬(n→∞)|e * [tex](n!) * (x+2)^n / ((n+1)! * (x+2)^(n+1))|[/tex]
= lim┬(n→∞)|e / (n+1) * (x+2)|
Taking the limit as n approaches infinity, we have:
lim┬(n→∞)|e / (n+1) * (x+2)| = 0
Therefore, the ratio simplifies to 0. Since the limit is less than 1, the series converges for all values of x. Hence, the radius of convergence is infinity, and the interval of convergence is (-∞, +∞).
To know more about radius of convergence,
https://brainly.com/question/2289050
#SPJ11
Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve y=xe −x
,2≤x≤4 (a) about the x-axis. ∫ 2
4
2π 1+e −2x
(1−x) 2
dx
∫ 2
4
2πy 1+e −2y
(1−y) 2
dy
∫ 2
4
2π 1+e −2y
(1−y) 2
dy
∫ 2
4
2πx 1+e −2x
(1−x) 2
dx
∫ 2
4
2πxe −x
1+e −2x
(1−x) 2
dx
(b) about the y-axis. ∫ 2
4
2πx 1+e −2x
(1−x) 2
dx
∫ 2
4
2π 1+e −2y
(1−y) 2
dy
∫ 2
4
2πy 1+e −2y
(1−y) 2
dy
∫ 2
4
2πxe −x
1+e −2x
(1−x) 2
dx
∫ 2
4
2π 1+e −2x
(1−x) 2
dx
Previous
The integrals for the area of the surface obtained by rotating the curve `y=xe^-x, 2 ≤ x ≤ 4` about the x-axis and the y-axis are given by:∫_2^4〖2πxy dy〗 = 2π [3e^(4) - 2] / e^(6) and ∫_2^4〖2πxy dx〗 = 2π [2 - 14e^(-4)] / e^(4), respectively.
To set up an integral for the area of the surface obtained by rotating the curve `y=xe^-x, 2 ≤ x ≤ 4` about the x-axis, we need to follow the steps given below:
Consider an element of the curve, `y=xe^-x`, and rotate it about the x-axis. This will generate a surface in the shape of a disk with radius x and thickness `dy`. Therefore, the area of the surface element is given by:`
dA = 2πxy dy`, where `x` is the element's distance from the axis of revolution, and `y = xe^-x`.
To find the area of the entire surface, we need to add up the area of all the elements from `y=2` to `y=4`. Therefore, the integral for the area of the surface obtained by rotating the curve `y=xe^-x, 2 ≤ x ≤ 4` about the x-axis is given by
:∫_2^4〖2πxy dy〗
= ∫_2^4〖2πxe^(-x) xe^xdx〗
= 2π ∫_2^4〖x e^(-x) dx〗
= 2π ∫_2^4〖x d(-e^(-x))
= 2π [x(-e^(-x))|_2^4 - ∫_2^4
= (-e^(-x) dx)]
= 2π [(-4e^(-4) + 2e^(-2)) + e^(-2) - e^(-4)]
= 2π [-2e^(-4) + 3e^(-2)]
= 2π [3e^(4) - 2] / e^(6)
The integral of the area of the surface obtained by rotating the curve `y = xe^-x, 2 ≤ x ≤ 4` about the y-axis can be set up similarly. We need to consider an element of the curve, `y=xe^-x`, and rotate it about the y-axis.
This will generate a surface in the shape of a disk with a radius `y` and thickness `dx`. Therefore, the area of the surface element is given by:`
dA = 2πxy dx`, where `y` is the element's distance from the axis of revolution, and `y = xe^-x`.
To find the area of the entire surface, we need to add up the area of all the elements from `x=2` to `x=4`. Therefore, the integral for the area of the surface obtained by rotating the curve `y=xe^-x, 2 ≤ x ≤ 4` about the y-axis is given by:
= ∫_2^4〖2πxy dx〗
= ∫_2^4〖2πxe^(-x) xe^xdx〗
= 2π ∫_2^4▒〖xe^(-x) x dx〗
= 2π ∫_2^4▒〖x^2 d(-e^(-x))〗
= 2π [x^2(-e^(-x))|_2^4 - ∫_2^4(-2xe^(-x) dx)]
= 2π [(-16e^(-4) + 4e^(-2)) + (4e^(-2) - 2e^(-4))]
= 2π [2 - 14e^(-4)] / e^(4)
Therefore, the integrals for the area of the surface obtained by rotating the curve `y=xe^-x, 2 ≤ x ≤ 4` about the x-axis and the y-axis are given by:∫_2^4〖2πxy dy〗 = 2π [3e^(4) - 2] / e^(6) and ∫_2^4〖2πxy dx〗 = 2π [2 - 14e^(-4)] / e^(4), respectively.
To know more about the integrals, visit:
brainly.com/question/14502499
#SPJ11
It is known that the Maclaurin series of function f(x)=tan x is tanx=x+ 1/3 x^3+ 2/15 x^5 +… Then the fifth derivative of the function g(x)=(2+x^2) tan x at 0 is g^(5)(0)=
The values of the given Maclaurin series for the function f(x) = tan x is found to be 0.
The given Maclaurin series for the function f(x) = tan x is:
tan x = x + (1/3)x³ + (2/15)x⁵ + ....
Given function is g(x) = (2 + x²) tan x
Using product rule, we can write:
g⁽⁵⁾(x) = (2 + x²) f⁽⁵⁾(x) + 10x³f³(x) + 15x²f²(x) + 10xf(x) + f(x)d⁽⁵⁾(2 + x²)/dx⁵
For finding g⁽⁵⁾(0), we need to find f⁽⁵⁾(0), f³(0), f²(0) and f(0) that are as follows:
f⁽⁵⁾(x) = 16(16x⁴ - 20x² + 5)tan x + 16x³sec² x (8x⁴ - 8x² + 1) + 480x⁴tan xf⁽⁵⁾(0)
= 16(0 - 20(0) + 5)(0) + 16(0)sec² 0(8(0) - 8(0) + 1) + 480(0)tan 0
= 0f³(x) = 2(2x² - 1)tan x + 4x(2 + x²)sec² xf³(0)
= 2(2(0)² - 1)(0) + 4(0)(2 + (0)²)sec² 0
= 0f²(x) = 2x(3 + x²)sec² xf²(0) = 2(0)(3 + (0)²)sec² 0
= 0f(x) = tan xf(0) = tan 0 = 0
Now,
d⁽⁵⁾(2 + x²)/dx⁵ = 0g⁽⁵⁾(0)
= (2 + (0)²)f⁽⁵⁾(0) + 10(0)³f³(0) + 15(0)²f²(0) + 10(0)f(0) + f(0)d⁽⁵⁾(2 + x²)/dx⁵
= (2)(0) + (0) + (0) + (0) + (0)
= 0
Know more about the Maclaurin series
https://brainly.com/question/28170689
#SPJ11
What is true about the completely simplified sum of the polynomials 3x2y2 − 2xy5 and −3x2y2 + 3x4y?
The sum is a trinomial with a degree of 5.
The sum is a trinomial with a degree of 6.
The sum is a binomial with a degree of 5.
The sum is a binomial with a degree of 6.
Answer:
The sum is a binomial with a degree of 5.
Answer:
D
Step-by-step explanation:
;)
Find the area of the region under the given curve from 1 to 3 . y=4x−x2x2+8
The resulting value will represent the area of the region under the curve y = (4x - x^2) / (x^2 + 8) from x = 1 to x = 3.
To find the area of the region under the curve y = (4x - x^2) / (x^2 + 8) from 1 to 3, we need to integrate the function over that interval.
The integral can be written as:
∫[1 to 3] (4x - x^2) / (x^2 + 8) dx
To evaluate this integral, you can use various integration techniques such as substitution or partial fractions. After performing the integration, you will obtain a numerical value.
The resulting value will represent the area of the region under the curve y = (4x - x^2) / (x^2 + 8) from x = 1 to x = 3.
To know more about area of curve, visit:
https://brainly.com/question/31024383
#SPJ11
Given the demand function q= sqrt(2500 - 2p^2) with domain [0, 25 sqrt(2)], determine
(a) the elasticity of demand E;
(b) the elasticity when p = 20 and interpret your results;
(c) the range of prices corresponding to elastic, unitary, and inelastic demand; and
(d) the range of quantities corresponding to elastic, unitary, and inelastic demand.
a) Elasticity of demand E:The elasticity of demand is a measure of the responsiveness of the amount of goods or services demanded to a change in price. It can be calculated using the formula:E = (p/q) * (dq/dp)where p is the price, q is the quantity demanded, and dq/dp is the derivative of q with respect to p.
we get:E = -2p^2 / (2500 - 2p^2) * 1 / sqrt(2500 - 2p^2)Multiplying the numerator and denominator by (2500 - 2p^2), we get:E = -2p^2 / (2500 - 2p^2)^3/2Therefore, the elasticity of demand is:E = -2p^2 / (2500 - 2p^2)^3/2b) Elasticity when p = 20 and interpretation:
The elasticity of demand when p = 20 can be found by substituting p = 20 into the elasticity formula:E = -2(20)^2 / (2500 - 2(20)^2)^3/2E = -800 / 1000E = -0.8Since the elasticity is negative, the demand is price inelastic. This means that a change in price will have a relatively small effect on the quantity demanded. Interpretation: If the price of the good increases, the quantity demanded will decrease, but not by a large amount.
We graph the function y = sqrt(2500 - 2x^2) and look for the ranges where the slope of the graph is greater than 1, equal to 1, and less than 1. We find that:Elastic demand: q > 500Unitary demand: q = 500Inelastic demand: q < 500Therefore, the ranges of quantities corresponding to elastic, unitary, and inelastic demand are: Elastic demand: q > 500 Unitary demand: q = 500 Inelastic demand: q < 500
To know more about demand visit:
https://brainly.com/question/30402955
#SPJ11
Let A be a symmetric n x n matrix. a) Show that if 7,7 € R", then (Au)-v=ū- (Av). b) Show that if A has distinct eigenvalues A₁ and A2, with corresponding eigenvectors v₁ and 2, then v₁ and 1₂ are orthogonal.
a. Its true that (Au)^-1 = u^-1 - (Av).
b. The v₁ and v₂ are orthogonal since their dot product is zero.
a) To show that (Au)^-1 = u^-1 - (Av), we can start by expressing u^-1 and v in terms of the eigenvectors of A.
Since 7 is an eigenvalue of A, there exists an eigenvector u corresponding to 7, i.e., Au = 7u.
Similarly, let's assume 1 is an eigenvalue of A with eigenvector v, i.e., Av = 1v.
Now, let's calculate (Au)^-1:
(Au)^-1 = (7u)^-1 [Substituting Au = 7u]
= 1/7 * u^-1
On the other hand, let's calculate u^-1 - (Av):
u^-1 - (Av) = u^-1 - (1v)
= u^-1 - v
Now, to prove that (Au)^-1 = u^-1 - (Av), we need to show that 1/7 * u^-1 = u^-1 - v.
We can rearrange the equation:
1/7 * u^-1 = u^-1 - v
Multiply both sides by 7:
u^-1 = 7u^-1 - 7v
Add 7v to both sides:
u^-1 + 7v = 7u^-1
Now, we can substitute Au = 7u:
u^-1 + 7v = A(u^-1)
Since A is a symmetric matrix, we can say that A(u^-1) is equivalent to (u^-1)A:
u^-1 + 7v = (u^-1)A
Therefore, we have shown that (Au)^-1 = u^-1 - (Av).
b) To show that v₁ and v₂ are orthogonal, we need to prove that their dot product is zero.
Let A₁ be an eigenvalue of A with eigenvector v₁:
Av₁ = A₁v₁
Similarly, let A₂ be another eigenvalue of A with eigenvector v₂:
Av₂ = A₂v₂
To show that v₁ and v₂ are orthogonal, we need to prove v₁ ⋅ v₂ = 0.
Taking the dot product of both sides of the equations above:
v₁ ⋅ (Av₂) = v₁ ⋅ (A₂v₂)
(Av₁) ⋅ v₂ = A₂(v₁ ⋅ v₂)
Since A is a symmetric matrix, Av₁ is equivalent to v₁A:
(v₁A) ⋅ v₂ = A₂(v₁ ⋅ v₂)
Expanding the dot product:
v₁ ⋅ (A ⋅ v₂) = A₂(v₁ ⋅ v₂)
Since A is symmetric, A ⋅ v₂ is equivalent to v₂A:
v₁ ⋅ (v₂A) = A₂(v₁ ⋅ v₂)
Again, expanding the dot product:
(v₁ ⋅ v₂)A = A₂(v₁ ⋅ v₂)
Since A₂ is a distinct eigenvalue, it is not equal to zero (A₁ ≠ A₂). Therefore, we can divide both sides of the equation by (A₂ - A₁):
(v₁ ⋅ v₂) = 0
Hence, we have shown that v₁ and v₂ are orthogonal since their dot product is zero.
Learn more about matrix at https://brainly.com/question/30358024
#SPJ11
Which of the following sets of vectors in R 3
are linearly dependent? Note. Mark all your choices. (0,8,4),(0,48,24)
(3,0,5),(4,−7,3),(8,1,6)
(8,0,4),(−9,8,5),(2,−2,6),(4,−4,0)
(3,8,0),(18,9,0)
The sets of vectors which are linearly dependent are:
(0,8,4),(0,48,24) and
(8,0,4),(−9,8,5),(2,−2,6),(4,−4,0)
and the sets of vectors which are linearly independent are:
(3,0,5),(4,−7,3),(8,1,6) and
(3,8,0),(18,9,0).
The set of vectors (0,8,4),(0,48,24) are linearly dependent because the second vector is exactly six times the first vector.
The set of vectors (3,0,5),(4,−7,3),(8,1,6) are linearly independent. We can see this by trying to solve the equation a(3,0,5) + b(4,-7,3) + c(8,1,6) = (0,0,0). This leads to the system of equations:
3a + 4b + 8c = 0
-7b + c = 0
5a + 3b + 6c = 0
Solving this system gives us a unique solution a=b=c=0 which means that the vectors are linearly independent.
The set of vectors (8,0,4),(−9,8,5),(2,−2,6),(4,−4,0) are linearly dependent because we can see that the fourth vector is the sum of the first three.
The set of vectors (3,8,0),(18,9,0) are linearly dependent because the second vector is exactly six times the first vector.
Therefore, the sets of vectors which are linearly dependent are:
(0,8,4),(0,48,24) and
(8,0,4),(−9,8,5),(2,−2,6),(4,−4,0)
and the sets of vectors which are linearly independent are:
(3,0,5),(4,−7,3),(8,1,6) and
(3,8,0),(18,9,0).
Learn more about vectors here:
https://brainly.com/question/24256726
#SPJ11
angelina drove at an average rate of 80 kph and then stopped 20 minutes for gas. after the stop, she drove at an average rate of 100 kph. altogether she drove 250 km in a total trip time of 3 hours including the stop. which equation could be used to solve for the time $t$ in hours that she drove before her stop?
Angelina drove for 0.83 hours (or approximately 50 minutes) before her stop.
The equation that could be used to solve for the time $t$ in hours that Angelina drove before her stop is:
$80t + 100(3 - t - \frac{1}{3}) = 250$
Let's break down the information given. Angelina drove at an average rate of 80 kph for a certain amount of time, which we want to find. After that, she stopped for 20 minutes (or $\frac{1}{3}$ of an hour) for gas. Then, she continued driving at an average rate of 100 kph. The total trip time, including the stop, was 3 hours.
To solve for the time Angelina drove before her stop, we can set up an equation based on the distance she traveled. The distance traveled at 80 kph is given by $80t$, where $t$ represents the time in hours. The distance traveled after the stop at 100 kph is $100(3 - t - \frac{1}{3})$, where $3 - t - \frac{1}{3}$ represents the remaining time after the stop.
The sum of these distances should equal the total distance traveled, which is 250 km. Therefore, we set up the equation $80t + 100(3 - t - \frac{1}{3}) = 250$.
By solving this equation, we can find the value of $t$, which represents the time in hours that Angelina drove before her stop.
To solve the equation, we can start by simplifying the expression on the right side:
$80t + 100(3 - t - \frac{1}{3}) = 250$
First, we can simplify the expression $3 - t - \frac{1}{3}$:
$3 - t - \frac{1}{3} = 2\frac{2}{3} - t = \frac{8}{3} - t$
Now, we substitute this expression back into the equation:
$80t + 100(\frac{8}{3} - t) = 250$
Next, we distribute the 100 to both terms inside the parentheses:
$80t + \frac{800}{3} - 100t = 250$
Combining like terms:
$-20t + \frac{800}{3} = 250$
To isolate the variable $t$, we can subtract $\frac{800}{3}$ from both sides:
$-20t = 250 - \frac{800}{3}$
To simplify the right side, we need a common denominator for 250 and $\frac{800}{3}$, which is 3:
$-20t = \frac{750}{3} - \frac{800}{3}$
Subtracting the fractions:
$-20t = \frac{-50}{3}$
Finally, we divide both sides by -20 to solve for $t$:
$t = \frac{\frac{-50}{3}}{-20} = \frac{50}{60} = \frac{5}{6}$
Therefore, Angelina drove for $\frac{5}{6}$ or 0.83 hours (or approximately 50 minutes) before her stop.
Learn more about approximately here:
brainly.com/question/31695967
#SPJ11
Which one of the following best describe Chi-square Distribution? Marked out of 100 Select one Flag question a symmetrically distributed. b. Positive skewness c. left-skewed distribution d. left-skewed and it can have negative possible values
Chi-Square Distribution is a symmetrically distributed, non-negative distribution that is commonly used to measure the degree of fit between the expected and the observed frequency of events. This distribution is particularly useful when working with categorical data, such as data from surveys or experiments, where outcomes can be classified into discrete categories.
Chisquare is the most commonly used distribution in the statistical analysis of discrete data. It was first introduced by Pearson and is widely used in hypothesis testing and model fitting. Chi-square distribution is an important concept in the field of statistical analysis and is used to calculate the significance of results from various statistical tests.
For example, it can be used to determine whether the difference between two sets of data is statistically significant or whether a particular relationship exists between two variables.
In summary, the best way to describe Chi-square Distribution is as a symmetrically distributed, non-negative distribution that is commonly used to measure the degree of fit between the expected and the observed frequency of events.
To know more about symmetrically visit:
https://brainly.com/question/31184447
#SPJ11
How many solutions will the linear system Ax = b have if b is in the column space of A and the column vectors of A are linearly dependent? Explain.
If the column vectors of matrix A are linearly dependent and vector b is in the column space of A, then the linear system Ax = b will have infinitely many solutions.
In a linear system Ax = b, where A is an m x n matrix, x is an n x 1 vector of variables, and b is an m x 1 vector, the column space of A represents all possible linear combinations of the column vectors of A.
If the column vectors of A are linearly dependent, it means that at least one column vector can be expressed as a linear combination of the other column vectors.
In other words, there exists a non-zero solution to the equation Ax = 0.
Now, if b is in the column space of A, it means that b can be expressed as a linear combination of the column vectors of A.
Since the column vectors of A are linearly dependent, we can find a non-zero solution to the equation Ax = b.
This means that there exists more than one solution to the linear system.
To see why there are infinitely many solutions, consider the equation Ax = b with a non-zero solution x₀.
Since the column vectors of A are linearly dependent, we can add a multiple of any column vector to another column vector without changing the column space of A.
Therefore, if x is a solution to the equation Ax = b, then x + kx₀ (where k is any scalar) will also be a solution.
This implies that there are infinitely many solutions to the linear system.
In conclusion, if the column vectors of matrix A are linearly dependent and vector b is in the column space of A, the linear system Ax = b will have infinitely many solutions.
Learn more about Vector here:
https://brainly.com/question/29740341
#SPJ11
If
cos(θ) = 1/9 and θ is in the 4th quadrant, find sin(θ)
If \( \cos (\theta)=\frac{1}{9} \) and \( \theta \) is in the 4 th quadrant, find \( \sin (\theta) \) \[ \sin (\theta)= \]
The value of sin(θ) is - (4/3)√5 when cos(θ) = 1/9 and θ is in the fourth quadrant.
Given the value of cosθ=1/9 and θ is in the 4th quadrant. We have to find the value of sinθ.
Let us try to plot it in the fourth quadrant.
Since the value of cosine is positive in the fourth quadrant, we have drawn an angle making an acute angle with the negative direction of x-axis. Now, we can use Pythagorean identity as:
cos²θ + sin²θ = 1
sin²θ = 1 - cos²θ
sinθ = ±√(1 - cos²θ)
Since the angle is in the fourth quadrant, the value of sinθ is negative. Hence, sinθ = - √(1 - (1/9)²)
Now, simplify it. We get:
sinθ = - √(80/81)
sinθ = - √80/9
sinθ = - (4/3)√5
Thus, the value of sin(θ) is - (4/3)√5 when cos(θ) = 1/9 and θ is in the fourth quadrant.
To know more about quadrant visit
https://brainly.com/question/3267281
#SPJ11
Use the limit definition of a derivative to find the derivative of g(x)=(x−1)2+1
The derivative of the function \(g(x) = (x-1)^2 + 1\) is \(g'(x) = 2x + 2\).
To find the derivative of the function \(g(x) = (x-1)^2 + 1\) using the limit definition of a derivative, we'll follow these steps:
Step 1: Write the limit definition of a derivative:
\[f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}\]
Step 2: Substitute the function \(g(x) = (x-1)^2 + 1\) into the limit definition:
\[g'(x) = \lim_{{h \to 0}} \frac{{[(x+h-1)^2 + 1] - [(x-1)^2 + 1]}}{h}\]
Step 3: Simplify the expression inside the limit:
\[g'(x) = \lim_{{h \to 0}} \frac{{(x^2 + 2xh + h^2 - 2x + 2h) - (x^2 - 2x + 1)}}{h}\]
Step 4: Combine like terms:
\[g'(x) = \lim_{{h \to 0}} \frac{{2xh + h^2 + 2h}}{h}\]
Step 5: Factor out \(h\) from the numerator:
\[g'(x) = \lim_{{h \to 0}} \frac{{h(2x + h + 2)}}{h}\]
Step 6: Cancel out the common factor \(h\) in the numerator and denominator:
\[g'(x) = \lim_{{h \to 0}} (2x + h + 2)\]
Step 7: Evaluate the limit as \(h\) approaches 0:
\[g'(x) = 2x + 0 + 2\]
Step 8: Simplify the expression:
\[g'(x) = 2x + 2\]
Therefore, the derivative of the function \(g(x) = (x-1)^2 + 1\) is \(g'(x) = 2x + 2\).
Learn more about derivative here
https://brainly.com/question/31399608
#SPJ11
Find The Slope Of The Line Tangent To The Polar Curve R=4cosθ At The Point Θ=−32π. Write The Exact Answer. Do Not Round.
The slope of the line tangent to the polar curve r = 4cosθ at the point θ = -32π is 0.
To find the slope of the line tangent to the polar curve r = 4cosθ at the point θ = -32π, we need to differentiate the polar equation with respect to θ and evaluate it at the given point.
The polar curve r = 4cosθ represents a circle with a radius of 4 centered at the origin.
To differentiate r = 4cosθ, we use the chain rule. The derivative of r with respect to θ is given by dr/dθ = -4sinθ.
Now, we can evaluate the derivative at θ = -32π:
dr/dθ = -4sin(-32π) = -4sin(-π) = -4(0) = 0
The slope of the line tangent to the polar curve at the point θ = -32π is equal to the derivative dr/dθ evaluated at that point. In this case, the slope is 0.
Therefore, the slope of the line tangent to the polar curve r = 4cosθ at the point θ = -32π is 0.
Learn more about slope here
https://brainly.com/question/16949303
#SPJ11
Let H be a group and K a group with an action on H; i.e. with a specified homomorphism of K into the automorphism group of H. 24 Finite Groups Denote the image of an element h of H under the element k of K by h k
. Let G be the set of formal products kh with h∈H and k∈K, where 1 K
⋅h is taken to be h and k⋅1 H
is taken to be k. Define a product on G by setting kh⋅k ′
h ′
=(kk ′
)(h k ′
h ′
) for h and h ′
in H and k and k ′
in K. Prove that G becomes a group, H a normal subgroup of G and K a subgroup of G. Also G=KH=HK, H∩K=⟨1⟩ and h k
is just the conjugate k −1
hk of h by k in G. (This group G is called the split extension or semidirect product of H by K.)
To prove that G is a group, we need to show that it satisfies the group axioms of closure, associativity, identity, and inverse.
Closure: For any kh and k′h′ in G, their product kh⋅k′h′ is also in G.
Associativity: The product operation in G is associative: (kh⋅k′h′)⋅k″h″ = kh⋅(k′h′⋅k″h″) for any kh, k′h′, and k″h″ in G.
Identity: The element 1K⋅1H serves as the identity element in G.
Inverse: For any kh in G, its inverse is given by k−1⋅h−1.
H is a normal subgroup of G because for any kh in G and any h′ in H, the conjugate k−1⋅hk is also in G.
K is a subgroup of G since it is closed under the product operation and has the identity element.
G = KH = HK, which means that every element in G can be expressed as a product of an element in K and an element in H.
H∩K = ⟨1⟩, meaning the intersection of H and K is the trivial subgroup consisting only of the identity element.
The expression hk represents the conjugate of h by k in G, denoted as k−1⋅hk.
"You writing needs to be legible and follow the format of the
question. You should have three answers clearly defined:
a) r(t) =
b) r(t) =
c) r(t) =
Please note this is ONE question.
Results for this submission Entered (b) If 7(4) = P and (8) = Q, then r(t) = (c) If the points P and Q correspond to the parameter values t = 0 and t = -4, respectively, then r(t) = Preview My Answer"
The answers are :a) r(t) = r0 + v0t + 1/2at²b) r(t)
= P + (Q - P)/8t² - Pc) r(t)
= (63/64)P + (1/64)Q
To solve the given question, let us start by using the formula for the position of an object moving with constant acceleration a from the position r0 with initial velocity v0 at time t.
The formula is given as; r(t) = r0 + v0t + 1/2at²(a) r(t)
= r0 + v0t + 1/2at²
Where r0 is the initial position of the object, v0 is its initial velocity, t is the time
for which we want to find the position, and a is the constant acceleration of the object.
(b) If 7(4) = P and (8) = Q,
then r(t) = We are given that 7(4) = P and (8) = Q.
Substituting these values into the formula above,
we get; P = r(4)Q = r(8)
We need to find r(t) using the values of P, Q, and t.
To do this, we will find an expression for r(t) in terms of P, Q, and t.
To eliminate r0 and v0 from the formula above, we can use the formula for the average velocity of an object over an interval of time.
The formula is given as; vave = (v0 + v)/2
where v0 is the initial velocity of the object, v is its final velocity, and vave is the average velocity of the object over the interval of time.
We can rearrange this formula to obtain an expression for v in terms of v0 and vave as follows; v = 2vave - v0
We can then substitute this expression for v into the formula for r(t) to obtain an expression for r(t) in terms of r0, v0, a, and t as follows;
r(t) = r0 + (v0 + 1/2at)² - v0²/2a
(b) r(t) = P + (Q - P)/8t² - P
(c) If the points P and Q correspond to the parameter values t = 0 and t = -4,
respectively, then r(t) = To find r(t) using the given values of P, Q, and t,
we can substitute t = -4,
P = Q = r0 into the formula above.
Doing this gives; r(t) = r0 + v0t + 1/2at² r(-4)
= r0 + v0(-4) + 1/2a(-4)² r(-4)
= r0 - 4v0 - 8a
We can then substitute P = r0,
Q = r(-4), and t = -4 into the formula for r(t) in part (b) to obtain the answer in terms of P and Q as follows;
r(t) = P + (Q - P)/8t² - P r(-4)
= P + (Q - P)/8(-4)² - P r(-4)
= P + (Q - P)/64 - P r(-4)
= (63/64)P + (1/64)Q
To know more about velocity visit:
https://brainly.com/question/30559316
#SPJ11
13) Find BC B A) 41 ft C) 38 ft 42° 109° с 21 ft B) 42 ft D) 44 ft
The diagram is a triangle ABC where angle
To find BC, we use the law of sines, which states that for any triangle ABC:
we can use the part of the formula that relates the sine of an angle to its opposite side:
We can then rearrange this formula to solve for BC:
To know more about triangle visit:-
brainly.com/question/2773823
#SPJ11
Suppose that \( \$ 15,548 \) is invested at an interest rate of \( 5.4 \% \) per year, compounded continuously. a) Find the exponential function that describes the amount in the account after time \(
Given: The initial amount is $15,548 and the interest rate is 5.4% compounded continuously.
We need to find the exponential function that describes the amount in the account after time.
Using the formula for continuous compounding, we can write the amount A(t) in the account after t years as follows:[tex]A(t) = P e^(r t)Where, P = principal amount (initial investment) = $15,548r = annual interest rate (as a decimal) = 5.4% = 0.054t = time in years[/tex]
[tex]Now, the exponential function that describes the amount in the account after time t is given by:A(t) = $15,548 e^(0.054t)[/tex]
Using the formula for continuous compounding.
[tex]Hence, the required exponential function is A(t) = $15,548 e^(0.054t).[/tex]
To know more about the word exponential visits :
https://brainly.com/question/3985777
#SPJ11
The exponential function is:
A = 15548 * e^(0.054t).
This equation describes the amount in the account after time t.
To find the exponential function that describes the amount in the account after time t, we can use the formula for continuous compound interest:
A = P * e^(rt),
where A is the amount in the account after time t, P is the principal amount, e is the base of the natural logarithm (approximately 2.71828), r is the interest rate, and t is the time in years.
In this case, the principal amount P is $15,548 and the interest rate r is 5.4% (or 0.054 as a decimal). Thus, the exponential function is:
A = 15548 * e^(0.054t).
This equation describes the amount in the account after time t.
To know more about exponential, visit:
https://brainly.com/question/29160729
#SPJ11
can you write for me a summary about chemical engineering working in distilled spirits career
Chemical engineering in the distilled spirits industry involves applying principles of chemistry and engineering to the production of alcoholic beverages.
Chemical engineers play a crucial role in the distilled spirits industry by designing and optimizing the processes involved in producing alcoholic beverages. They work on various aspects such as fermentation, distillation, and blending. In the fermentation stage, they ensure the conversion of sugars into alcohol by carefully selecting and controlling yeast strains and monitoring temperature and pH levels.
During distillation, chemical engineers optimize the separation of alcohol from impurities to achieve the desired flavor and alcohol content. They also work on blending different spirits to create unique flavors. Chemical engineers collaborate with microbiologists, quality control specialists, and production teams to ensure the highest quality and consistency of the final product. Their expertise in chemistry, thermodynamics, and process control enables them to develop efficient and sustainable processes in the distilled spirits industry.
Know more about Chemical engineering here:
https://brainly.com/question/30764981
#SPJ11
Find the area of this triangle
8
123 degrees
15
Answer:
50.32 square units
Step-by-step explanation:
To find the area of a triangle, given the measures of two of its side lengths and the included angle, use the Sine Rule.
[tex]\boxed{\begin{minipage}{6 cm}\underline{Sine Rule - Area of a triangle} \\\\$A=\dfrac{1}{2}ab \sin C$\\\\where:\\ \phantom{ww}$\bullet$\;\;$a$ and $b$ are the sides.\\ \phantom{ww}$\bullet$\;\;$C$ is the incl\:\!uded angle. \\\end{minipage}}[/tex]
From inspection of the given triangle:
a = 8b = 15C = 123°Substitute these values into the formula and solve for A:
[tex]\begin{aligned}A&=\dfrac{1}{2} \cdot 8 \cdot 15 \cdot \sin 123^{\circ}\\\\&=4 \cdot 15 \cdot \sin 123^{\circ}\\\\&=60 \cdot \sin 123^{\circ}\\\\&=50.3202340...\\\\&=50.32\; \sf square\;units\;(nearest\;hundredth)\end{aligned}[/tex]
Therefore, the area of the given triangle is 50.32 square units (rounded to the nearest hundredth).
Let X and Y be continuous random variables having a joint probability density function (pdf) given by f(x, y) = e-y, (i) (ii) (iii) = < y < x, 0 < x <[infinity]⁰. Calculate P(X + Y < 1). Derive the conditional pdf of Y given that X = x. Find the value of P(Y> 2|X = 3). (10 marks)
Previous question
To calculate P(X + Y < 1), we need to find the probability that the sum of X and Y is less than 1.
We start by considering the region in the x-y plane where X + Y < 1. This region is defined by the conditions y < 1 - x and 0 < x < ∞.
To find the probability, we integrate the joint probability density function (pdf) over this region:
P(X + Y < 1) = ∫∫(region) f(x, y) dA
where dA represents the area element.
The region can be visualized as the triangle below the line y = 1 - x in the first quadrant.
Integrating over this region, we have:
P(X + Y < 1) = ∫∫(region) e^(-y) dy dx
= ∫[0, ∞] ∫[0, 1 - x] e^(-y) dy dx
= ∫[0, ∞] [-e^(-y)]|[0, 1 - x] dx
= ∫[0, ∞] (-e^(1 - x) + 1) dx
= -e^(1 - x) + x|[0, ∞]
= (0 - (-e)) + (∞ - 0)
= 1 + ∞
Since the result is ∞, we say that P(X + Y < 1) diverges or is undefined.
Moving on to the second part of the question, we need to find the conditional probability density function (pdf) of Y given that X = x.
The conditional pdf of Y given X = x is given by:
f(y|x) = f(x, y) / fX(x)
where fX(x) is the marginal pdf of X.
To find fX(x), we need to integrate the joint pdf over the range of y:
fX(x) = ∫[0, x] e^(-y) dy
= -e^(-y)|[0, x]
= -e^(-x) + 1
Now we can calculate the conditional pdf:
f(y|x) = e^(-y) / (-e^(-x) + 1)
Finally, we can find the value of P(Y > 2 | X = 3) by integrating the conditional pdf over the range of y > 2, given X = 3:
P(Y > 2 | X = 3) = ∫[2, ∞] f(y|x=3) dy
= ∫[2, ∞] e^(-y) / (-e^(-3) + 1) dy
= -e^(-y) / (-e^(-3) + 1)|[2, ∞]
= (-e^(-∞) / (-e^(-3) + 1)) - (-e^(-2) / (-e^(-3) + 1))
= 0 - (-e^(-2) / (-e^(-3) + 1))
= e^(-2) / (-e^(-3) + 1)
the value of P(Y > 2 | X = 3) is e^(-2) / (-e^(-3) + 1).
To know more about probability visit:
brainly.com/question/31828911
#SPJ11
Suppose ∫ a
b
f exists and is positive. Prove that there exists a subinterval J⊂[a,b] and a constant m>0 such that f(x)≥m for x∈J. 10) Let f be continuous on [a,b]. Show that there is a number c in (a,b) with
It is proven the existence of a subinterval J ⊂ [a, b] and a constant m > 0 such that f(x) ≥ m for x ∈ J.
How to prove that existence of a subinterval J ⊂ [a, b] and a constant m > 0 such that f(x) ≥ m for x ∈ J.To prove that there exists a subinterval J ⊂ [a, b] and a constant m > 0 such that f(x) ≥ m for x ∈ J, given that ∫[a,b] f(x) dx > 0, we can use the properties of continuous functions and the definition of the integral.
Because f(x) is continuous on the interval [a, b], its minimum and maximum values are found on this interval. We'll call the minimal value f min and the highest value f max.
Because the integral [a,b] f(x) dx > 0, the function f(x) is not always zero on the interval [a, b]. As a result, there exists at least one point x0 in [a, b] where f(x0) > 0.
Consider the subinterval J = [x0 -, x0 + ] [a, b], where is a positive value small enough to ensure that J remains within the interval [a, b].
f(x) is also continuous on J because it is continuous on [a, b]. The Extreme Value Theorem states that f(x) has a minimum value m on the closed and limited interval J. It is worth noting that m > 0 since f(x0) > 0.
Therefore, we have proven the existence of a subinterval J ⊂ [a, b] and a constant m > 0 such that f(x) ≥ m for x ∈ J.
Learn more about function at https://brainly.com/question/11624077
#SPJ4
