**Probability**

Question a. Let X be a Bernoulli random variable with P(X = 0) = I — p, P(X = I)= p. Let Y = I — X and Z = XY. (a) (5 points) Fuld P(X = x, Y = y) for x, y, e (0, (b) points) Find P(X = x,Z = z) for x, y,a a {0, 2. Question so. X and Y are independent Poisson random variables with perm- mess a and

(a) (5 points) Show that X+ Y has a Poisson distribution with parameter a +ft

(b) (5 points) Show that the conditional distribution of X, given X, Y = n is binomial and fmd its parameMts.

Question 23. Find the values of c to make the following fimetions valid probe- Nay density functions.

(a) (5 Paints) f(x)= cfgl —x)}-, for 0< x< I (Hint the derivative of sin-‘(x)= v.,=)

(b) (5 points) f(x). ceop(—x — for x e R

(5 Points) f(x)= C(5.x’ — coa(x)) for 00 x51

Question X is a uniform random variable distributed on [0, ‘xJ. Find the pdf of Y = sinX.

Question .5. Xis a continuous random variable with pdf, x >0

fx(x) = 0, x 5 0

(a) (4 Paints) Find EIX1

(b) (3 points) Find EIX1 (c) (3 points) Find Var[X]