Given joint density function for X and Y are given
To find probability density function for Z= X+Y.
Problem 3 (15pts). Let X and Y have joint pdf 0. else Find the pdf of...
Problem 5 Let X and Y be random variables with joint PDF Px.y. Let ZX2Y2 and tan-1 (Y/X). Θ i. Find the joint PDF of Z and Θ in terms of the joint PDF of X and Y ii. Find the joint PDF of Z and Θ if X and Y are independent standard normal random variables. What kind of random variables are Z and Θ? Are they independent?
Problem 5 Let X and Y be random variables with joint...
Problem 3 Let X and Y have joint pdf: fxy(x, y) = k(x + y) for 0 sxs1,0 s y s 1. (a) Find k. (b) Find the joint cdf of (X, Y). (c) Find the marginal pdf of X and of Y. (d) Find P[X < Y), P[Y < X²), P[X + Y > 0.5). (a) Find E[(X + Y)?]. (b) Find the variance of X + Y. (c) Under what condition is the variance of the sum equal...
Let X and Y have the joint pdf f(x,y) = e-x-y I(x > 0,y > 0). a. What are the marginal pdfs of X and Y ? Are X and Y independent? Why? b. Please calculate the cumulative distribution functions for X and Y, that is, find F(x) and F(y). c. Let Z = max(X,Y), please compute P(Z ≤ a) = P(max(X,Y) ≤ a) for a > 0. Then compute the pdf of Z.
Let (X,Y) have joint pdf given by I c, \y < x, 0 < x < 1, f(x, y) = { | 0, 0.W., (a) Find the constant c. (b) Find fx(r) and fy(y) (c) For 0 < x < 1, find fy\X=z(y) and HY|X=r and oſ X=z- (d) Find Cov(X, Y). (e) Are X and Y independent? Explain why.
Problem 2 (15pts). Consider the following joint density function 0, else (a) Find the conditional density function of Y given X (b) Find E(Y|X). (c) Find Var(Y|x).
Let X and Y have joint pdf f(x, y)= e if 0 < x < y< o and zero otherwise. Find Е(X |у). 16.
Let X and Y have the joint pdf fXY(x,y) = 24xy^3, 0<y<x<1. Find P(X>2/3, Y<1/3) Find P(X<2Y)
3. (50 pts) Let (X, Y) have joint pdf given by c, y x, 0 < x < 1, f(x, y) 0, o.w., (a) Find the constant c. (b) Find fx(x) and fy (y) (c) For 0 < 1, find fyx=x(y) and pyjx=x and oy Y|X=x (d) Find Cov(X, Y) (e) Are X and Y independent? Explain why.
. 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
3. (50 pts) Let (X,Y) have joint pdf given by -{ c, lyl< x, 0 < x < 1, f(x,y) = 0, 0.w., (a) Find the constant c. (b) Find fx(x) and fy(y) (c) For 0< x<1, find fy x-() and pyix- and ox (d) Find Cov(X, Y) (e) Are X and Y independent? Explain why