E = "Expected Value"
V = "Variance"
Problem 5. The joint density of X and Y is given by e" (z+y) fx.-otherwise. İf 0 < x < oo, 0 < y < 00, Consider the random variable Z-; a) Find the cumulative distribution function of Z b) What is the probability density function of Z?
1. Verify that fxr (x,y) -2e-x-y 0 < x < 00, x < y < is joint probability density function 2. Compute the probability that X < 1.and Y < 2.
Consider fx (x)=e*, 0<x and joint probability density function fx (x, y) = e) for 0<x<y. Determine the following: (a) Conditional probability distribution of Y given X =1. (b) ECY X = 1) = (c) P(Y <2 X = 1) = (d) Conditional probability distribution of X given Y = 4.
Suppose that X and Y are jointly continuous random variables with joint probability density function f(x,y) = {12rºy, 1 0, 0<x<a, 0<y<1 otherwise i) Determine the constant a ii) Find P(0<x<0.5, O Y<0.25) HE) Find the marginal PDFs fex) and y) iv) Find the expected value of X and Y. Le. E(X) and E(Y) v) Are X and Y independent? Justify your answer.
(4) Suppose that the joint density function of X, Y and Z is given by )<y <<< 1 f(x, y, z) = { otherwise. (a) Find the marginal density fz(z) (b) Find the marginalized density fxy(x, y) 72 (c) Find E (2)
# 6 If two random variables have the joint density f(x, y)=59 y?) for 0<x<1, 0<y<1 0 elsewhere a. Find the probability that 0.2 X<0.5 and 0.4<Y<0.6. b. Find the probability distribution function F(x, y). c. Are x and y independent?
Problem 4 Determine the value of c that makes the function: fry(x,y)s cry for 0 < x < 2 and 0 < y < 1 a valid joint probability density function. Determine the following: (c) P(X 1, Y> 0) (d) Marginal probability distributions of X and Y. What is the relationship between these random variables (e) P(Y X-1)
2. Let X and Y have joint density f(x.v) = \ şcy? if 0 <x< 1 and 1 <y<2, otherwise. (a) Compute the marginal probability density function of Y. If it's equal to 0 outside of some range, be sure to make this clear. (b) Set up but do not compute an integral to find P(Y < 2X).
8), Let X and Y be continuous random variables with joint density function f(x,y)-4xy for 0 < x < y < 1 Otherwise What is the joint density of U and V Y
11. The joint density function of X and Y is given by le(s+u) 0<x< o0,0<y<0 fla,y) = 01 %3D otherwise Find the density function of the random variable [Hint Use the distribution function of