The pdf of random variables is
The region is the triangle in the first quadrant between whose area is . Hence
The marginal pdfs are
Similarly,
a) We can see that , are not independent.
b) The probability . Since the region is square of area 0.25.
Similarly,
We can see that , are not conditionally independent given .
c) The expected value
Random variables z and y described by the PDF if x-+ yo 1 and x.> 0...
Let X and Y be continuous random variables with following joint pdf f(x, y): y 0<1 and 0<y< 1 0 otherwise f(x,y) = Using the distribution method, find the pdf of Z = XY.
The random variables X and Y have the joint PDF fx,y(x,y)=0.5, if x>0 and y>0 and xtys2, and 0 otherwise. Let A be the event Ys1) and let B be the event (Y>X). (You can use rational numbers like 3/5 for your answers.) 1. Calculate P(BIA). 2. Calculate fxıy(xlO.9) fxIY(0.39820710.9) 3. Calculate the conditional expectation of X, given that Y=1.8 4, Calculate the conditional variance of X, given that Y=1.4 5. Calculate fxlB(x) fXIB(0.11) 6. Calculate E[XY]. 7. Calculate the...
Calculate the following for the random vector (XY) with joint pdf fixy)--(3/4)(x+y) if 2x<yco, -1<x<o. 1. The marginal pdf of X and the marginal pdf of Y. Are X and Y independent random variables? 2. The expected value and variance of X and Y respectively. 3. The joint cdf in the case 2x<y<0. -1<x<0. 4. The expected value of the random variable Z defined as X^2 times Y^2. 5. The covariance between X and Y. 6. The expected value and...
Let X and Y be independent random variables with pdf 2-y , 0sys2 2 f(x) 0, otherwise 0, otherwise ) Find E(XY) b) Find Var (2X+3Y)
1. Consider a pair of random variables (X, Y) with joint PDF fx,y(x, y) 0, otherwise. (a) 1 pt - Find the marginal PDF of X and the marginal PDF of Y. (b) 0.5 pt - Are X and Y independent? Why? (e) 0.5 pt - Compute the mean of X and the mean of Y.
The random variables X and Y have the joint PDF -fa.. 2 0 S x s 1 0 Sy s1 (2х + Зу) fxy(x, y) = otherwise The mean squared error is defined as ET(X + Y - t)21, what value of t minimizes this error? The random variables X and Y have the joint PDF -fa.. 2 0 S x s 1 0 Sy s1 (2х + Зу) fxy(x, y) = otherwise The mean squared error is defined as...
The random variables X and Y have the joint PDF -fa.. 2 0 S x s 1 0 Sy s1 (2х + Зу) fxy(x, y) = otherwise The mean squared error is defined as ET(X + Y - t)21, what value of t minimizes this error? The random variables X and Y have the joint PDF -fa.. 2 0 S x s 1 0 Sy s1 (2х + Зу) fxy(x, y) = otherwise The mean squared error is defined as...
Question 3 [17 marks] The random variables X and Y are continuous, with joint pdf 0 y otherwise ce fxx (,y) a) Show that cye fr (y) otherwise and hence that c = 1. What is this pdf called? (b) Compute E (Y) and var Y; (c) Show that { > 0 fx (a) e otherwise (d) Are X and Y independent? Give reasons; (e) Show that 1 E(XIY 2 and hence show that E (XY) =. Question 3 [17...
5. (50pt) X and Y are continuous random variables with pdf f(x, y) 2r for 0 < x y < 1, and f(x,y) = 0 otherwise. Find the conditional expectation of Y given X = z.
1. Suppose X and Y are continuous random variables with joint pdf f(x,y) 4(z-xy) if = 0 < x < 1 and 0 < y < 1, and zero otherwise. (a) Find E(XY) b) Find E(X-Y) (c) Find Var(X - Y) (d) What is E(Y)?