![Problem 1A (expectation and p.d.f. of a function of a random variable, Y-g(X)). Consider a random variable X~Uniformle, e2] and define the new random variable Y - In X (a) Compute E(Y), using the theorem that states that E(g(()x() dz. (b) Now we will calculate E(Y) by computing the probability density function of Y first. To do so the initial step is to figure out what the c.d.f. of Y is, by noticing that: Note that in step ) we have applied the exponential function to both sides of the inequality, and used the fact that such function is monotone increasing (that is, a b if and only if e eb. Had we used a monotone decreasing function, we would have had to switch the direction of the inequality!). To find the p.d.f. of Y apply the chain rule2o Now that you have fy, compute the expectation of Y via the usual formula E(Y) = yfy(y) dy. You should get the same result as in part (a).](http://img.homeworklib.com/questions/5c50ddb0-6f76-11ea-b668-afaac05072e3.png?x-oss-process=image/resize,w_560)
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Problem 1A (expectation and p.d.f. of a function of a random...
1. Consider a continuous random variable X with the probability density function Sx(x) = 3<x<7, zero...
1. Consider a continuous random variable X with the probability density function Sx(x) = 3<x<7, zero elsewhere. a) Find the value of C that makes fx(x) a valid probability density function. b) Find the cumulative distribution function of X, Fx(x). "Hint”: To double-check your answer: should be Fx(3)=0, Fx(7)=1. 1. con (continued) Consider Y=g(x)- 20 100 X 2 + Find the support (the range of possible values) of the probability distribution of Y. d) Use part (b) and the c.d.f....
Q5. Simulations to estimate the expectation Let X be a Gaussian random variable with mean 0...
Q5. Simulations to estimate the expectation Let X be a Gaussian random variable with mean 0 and variance 1 i.e., h = 0 and o = 1). Use R code to take 10k samples from X. (a) plot the histogram and compare with the p.d.f. of X (using the formula from the textbook or Wikipedia). Show both plots. (b) compute E[X] empirically (i.e., for each sample compute X" and take their average); now repeat this computation with 50k samples.
Assume that the density function for a continuous random variable, Y, is defined as fY(y) =...
Assume that the density function for a continuous random variable, Y, is defined as fY(y) = 9y. exp(-3y) for (y>0) and f'(y) = 0 elsewhere. Given Y = y, the conditional C.D.F. for X is FX\Y (x\Y) =P[X 5 X Y = y) = 1 – exp(-x •y) for (x > 0). Questions below are related to the marginal distribution for X. 1. Derive the density, f* (x). 2. Evaluate the expectation, E(X)
Problem 4 Let X be a discrete random variable with probability mass function fx(x), and let...
Problem 4 Let X be a discrete random variable with probability mass function fx(x), and let t be a function. Define Y = t(X): that is, Y is the randon variable obtained by applying the function t to the value of X Transforming a random variable in this way is frequently done in statistics. In what follows, let R(X) denote the possible values of X and let R(Y) denote the possible values of To compute E[Y], we could irst find...
that E{E(Y|X) = E) (3 marks) If the random variable X has p.d.f. - SXSTE f(x)...
that E{E(Y|X) = E) (3 marks) If the random variable X has p.d.f. - SXSTE f(x) = {20 'o, otherwise, y = ex Gly)= Prob (Ys y) = Probe Prob(ancex) sluca inly) x < lncy F(x) dx = e cumulative distribution function technique to determine the p.d.f. of Y=e (4 marks CJE marks) avoy Given that the continuous random variable X and Y have joint p.d. f. f(x,y). She
6. A random variable Y has density function fy(a)Ky(where y 2 2 (and zero otherwise) and...
6. A random variable Y has density function fy(a)Ky(where y 2 2 (and zero otherwise) and b > 0. This random variable is obtained as the transformation Y-g(X) of the random variable X with density function fx(x) e, a 2 0. Function g(x) is an increasing function in r (a) Show that Kb2b. (b) Determine the transformation g(. in terms of b. Hint: For part (b), carefully read Wackerly 6.4 on how the method of transformations is derived. On p.311,...
Two random variables, X and Y, have joint probability density function f ( x , y...
Two random variables, X and Y, have joint probability density function f ( x , y ) = { c , x < y < x + 1 , 0 < x < 1 0 , o t h e r w i s e Find c value. What's the conditional p.d.f of Y given X = x, i.e., f Y ∣ X = x ( y ) ? Don't forget the support of Y. Find the conditional expectation E [...
Please show work :) Will upvote/rate! 4. Expectation of Product of Random Variables Proof From the definition of the ex...
Please show work :) Will upvote/rate!
4. Expectation of Product of Random Variables Proof From the definition of the expected value, the expected value of the product of two random variables is ı r P(X Y r2) E(X- Y) ri r2 where the sum is over all possible values of rı and r2 that the variable X and Y can take on (a) Using the definition above formally prove that if the events X = r1 and Y = r2...
Let the random variable X and Y have the joint probability density function. fxy(x,y) lo, 3....
Let the random variable X and Y
have the joint probability density function.
fxy(x,y) lo, 3. Let the random variables X and Y have the joint probability density function fxy(x, y) = 0<y<1, 0<x<y otherwise (a) Compute the joint expectation E(XY). (b) Compute the marginal expectations E(X) and E(Y). (c) Compute the covariance Cov(X,Y).
1.8. Expectation of a Random Variable 67 1.8.8. A bowl contains 10 chips, of which 8...
1.8. Expectation of a Random Variable 67 1.8.8. A bowl contains 10 chips, of which 8 are marked $2 each and 2 are $5 each. Let a person choose, at random and without replacement, three chips from this bowl. If the person is to receive the sum of the resulting amounts, find his expectation. Let f(z) = 2z, o < 1, zero elsewhere, be the pdf of X. (a) Compute E(1/X). (b) Find the odf and the pdf of Y...
1. Consider a continuous random variable X with the probability density function Sx(x) = 3<x<7, zero elsewhere. a) Find the value of C that makes fx(x) a valid probability density function. b) Find the cumulative distribution function of X, Fx(x). "Hint”: To double-check your answer: should be Fx(3)=0, Fx(7)=1. 1. con (continued) Consider Y=g(x)- 20 100 X 2 + Find the support (the range of possible values) of the probability distribution of Y. d) Use part (b) and the c.d.f....
Q5. Simulations to estimate the expectation Let X be a Gaussian random variable with mean 0 and variance 1 i.e., h = 0 and o = 1). Use R code to take 10k samples from X. (a) plot the histogram and compare with the p.d.f. of X (using the formula from the textbook or Wikipedia). Show both plots. (b) compute E[X] empirically (i.e., for each sample compute X" and take their average); now repeat this computation with 50k samples.
Assume that the density function for a continuous random variable, Y, is defined as fY(y) = 9y. exp(-3y) for (y>0) and f'(y) = 0 elsewhere. Given Y = y, the conditional C.D.F. for X is FX\Y (x\Y) =P[X 5 X Y = y) = 1 – exp(-x •y) for (x > 0). Questions below are related to the marginal distribution for X. 1. Derive the density, f* (x). 2. Evaluate the expectation, E(X)
Problem 4 Let X be a discrete random variable with probability mass function fx(x), and let t be a function. Define Y = t(X): that is, Y is the randon variable obtained by applying the function t to the value of X Transforming a random variable in this way is frequently done in statistics. In what follows, let R(X) denote the possible values of X and let R(Y) denote the possible values of To compute E[Y], we could irst find...
that E{E(Y|X) = E) (3 marks) If the random variable X has p.d.f. - SXSTE f(x) = {20 'o, otherwise, y = ex Gly)= Prob (Ys y) = Probe Prob(ancex) sluca inly) x < lncy F(x) dx = e cumulative distribution function technique to determine the p.d.f. of Y=e (4 marks CJE marks) avoy Given that the continuous random variable X and Y have joint p.d. f. f(x,y). She
6. A random variable Y has density function fy(a)Ky(where y 2 2 (and zero otherwise) and b > 0. This random variable is obtained as the transformation Y-g(X) of the random variable X with density function fx(x) e, a 2 0. Function g(x) is an increasing function in r (a) Show that Kb2b. (b) Determine the transformation g(. in terms of b. Hint: For part (b), carefully read Wackerly 6.4 on how the method of transformations is derived. On p.311,...
Please show work :) Will upvote/rate!
4. Expectation of Product of Random Variables Proof From the definition of the expected value, the expected value of the product of two random variables is ı r P(X Y r2) E(X- Y) ri r2 where the sum is over all possible values of rı and r2 that the variable X and Y can take on (a) Using the definition above formally prove that if the events X = r1 and Y = r2...
Let the random variable X and Y
have the joint probability density function.
fxy(x,y) lo, 3. Let the random variables X and Y have the joint probability density function fxy(x, y) = 0<y<1, 0<x<y otherwise (a) Compute the joint expectation E(XY). (b) Compute the marginal expectations E(X) and E(Y). (c) Compute the covariance Cov(X,Y).
1.8. Expectation of a Random Variable 67 1.8.8. A bowl contains 10 chips, of which 8 are marked $2 each and 2 are $5 each. Let a person choose, at random and without replacement, three chips from this bowl. If the person is to receive the sum of the resulting amounts, find his expectation. Let f(z) = 2z, o < 1, zero elsewhere, be the pdf of X. (a) Compute E(1/X). (b) Find the odf and the pdf of Y...
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