Mathematics Homework Help

4. Let X be the random variable with moment generating function

M(t) = e2t+50t2

(a) (5 points) Calculate P(−8 X 22).

(b) (5 points) Calculate P(|X − 2| >

p384.1).

5. Let X be a continuous random variable with probability density function

f(x) = 2x for 0 x 1.

Let N be a fixed nonnegative real number, and let Y be a random variable

given by

Y =

(

3N if X N;

4X − N if X < N.

(a) (10 points) Prove that E[Y ] = −4

3 N3 + 3N for N 1. (Explain

your reasoning clearly)

(b) (10 points) Prove that E[Y ] = 83

− N for N 1. (Explain your

reasoning clearly)

(c) (10 points) Find the value of N that maximizes E[Y ]. (Explain your

reasoning clearly.)