PSTAT 109 Lecture 13: Solving Random Variables

cr + k if x < 2 (c) If f(x) = { 22 + bx if x > 2 | , find the values — if any — of constants b, c, k such that f is differentiable and f'(1) = 3. ANSWER ONLY

Consider a binomial experiment with n=20 and p=0.70.

• Compute f(12).
• Compute f(16).
• Compute P(x >= 16).
• Compute P(x <= 15).
• Compute E(X).
• Compute Var(X).

The joint probability density function of X and Y is given by

f(x,y)={e^-(x+y)   for X>0, y>0      and    0   elsewhere

A. Find the marginal density of X

B. Find the marginal density of Y

C. Find the Conditional density of X given Y

D. Are random variables X and Y independent? State the reason of your answer.

E. Find P(X<.5, y<.5)

F. Find P(X=.5, y<.5)