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)
1 x Suppose X has an exponential distribution, thus its pdf is given by fx (x) = 5e8,0 5x<0, 2> 0;0 0.w. a. Find E(X) b. Find E(X(X-1) c. Find Var (x)
4. Suppose X and Y have the joint pdf f(x,y) = 6x, 0 < x < y < 1, and zero otherwise. (a) Find fx(x). (b) Find fy(y). (c) Find Corr(X,Y). (d) Find fy x(y|x). (e) Find E(Y|X). (f) Find Var(Y). (g) Find Var(E(Y|X)). (h) Find E (Var(Y|X)]. (i) Find the pdf of Y - X.
Let X, Y be jointly continuous with joint density function (pdf) fx,y(x, y) *(1+xy) 05 x <1,0 <2 0 otherwise (a) Find the marginal density functions (pdf) fx and fy. (b) Are X and Y independent? Why or why not?
3. Suppose X ~ Beta(a, β) with the constants α, β > 0, Define Y- 1-X. Find the pdf of Y
Consider the pair of random variables (X,Y). Suppose that marginally X ~ Binomial(2, ) and Y ~ Binomial(2, 3). If P(X > Y) = 0 and P(X = 0, Y = 2) = 16, then P(X = 1, Y = 1) equals
(10 points) Let Y have probability density function (pdf) 3y?, for ( <y<1 fy(y) = 10, otherwise (a) Compute the probability density function (pdf) of 1/Y. (b) Compute the probability density function (pdf) of Y1 +Y2, if Yį and Y2 are inde- pendent random variables with the same pdf as Y. (You can use a computer to help with the integration).
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)?
3. Let X has the following pdf: {. -1 <1 fx(a) otherwise 1. Find the pdf of U X2. 2. Find the pdf of W X
7. Suppose X and Y have joint pdf f(x,y) = 24x y if x >0, y>0,x+y<1 and 0 otherwise. Find P(Y > 2x).