
please explain each step 5.21 Let X and Y be independent random variables with fae-ax, x>0...
4.5.4 X and Y are random variables with the joint PDF ( 5x2/2 JX,Y (x, y) = -1 < x < 1; 0 <y < x2, otherwise. 10 (a) What is the marginal PDF fx(x)? (6) What is the marginal PDF fy(y)?
5.5.5 X and Y are random variables w the joint PDF X,Y (z, y) = 0 otherwise. (a) What is the marginal PDF fx()? (b) What is the marginal PDF fr(v)?
. For > 0 and A > 0, define the joint pdf -Ay = 0<x<A,<y, fx.y(,y) 10 else. (a) Express c in terms of X and A. (b) Find E[XY]. (c) Let [2] be the largest integer less than or equal to z. For example, (3.2] = 3 and [2] = 2. Find the probability that [Y] is even, given that 4 <x< 34
NIS 4) The joint pdf of X and Y is 1, 0<x<1, 0<y< 2x, fx,8(8,y) = { 0, otherwise. otherwise. or 1 (Note: This pdf is positive (having the value 1) on a triangular region in the first quadrant having area 1.) Give the cdf of V = min{X, Y}. x
The random Variable X has a pdf fx (2) = {*** kr + > -1 <r<2 otherwise Y is a function of X and is derived using Y = g(x) = X S -X X2 X <0 X>0 Find: (A) fr(y) (B) E[Y] using fy(y) (C) EY] using fx (2)
Let X and Y be random variables with joint PDF fx,y(x, y) = 2 for 0 < y < x < 1. Find Var(Y|X).
4. Let X and Y have joint probability density function f(x,y) = 139264 oray3 if 0 < x, y < 4 and y> 4-1, otherwise. (a) Set up but do not compute an integral to find E(XY). (b) Let fx() be the marginal probability density function of X. Set up but do not compute an integral to find fx(x) when I <r54. (c) Set up but do not compute an integral to find P(Y > X).
Let X and Y be continuous random variables with joint pdf f(x,y) =fX (c(X + Y), 0 < y < x <1 otBerwise a. Find c. b. Find the joint pdf of S = Y and T = XY. c. Find the marginal pdf of T. 、
(6 points) Let X and Y be independent random variables with p.d.f.s fx(x) -{ { 1-22 0, for |2|<1, otherwise. fy(y) = for y>0, otherwise. 0, Let W = XY (a) (2 points) Find the p.d.f. of W, fw(w). (b) (2 points) Find the moment generating function of W2, Mw?(t) = E (e«w?). (c) (2 points) Find the conditional expectation of W given Y = y, E(W|Y = y).
7. The random variables X and Y have joint probability density function f given by 1 for x > 0, |y| 0 otherwise. 1-x, Below you find a diagram highlighting the (r, y) pairs for which the pdf is 1 (a) Calculate the marginal probability density function fx of X (b) Calculate the marginal cumulative distribution function Fy of Y (c) Are X and Y independent? Explain.