IV Y is a discrete random variable with probability distribu

IV: Y is a discrete random variable with probability distribution function P(Y=y)= y

k

where y = 1,3,5,12,15.
1. Find the value of k that makes this a legitimate probability distribution.

2. Find E(Y)

3. Find the moment-generating function of Y

4. Using the moment-generating function from (3), find V(Y)

Solution

1.

Sum of all probabilities should be 1. So,

1K+3K+5K+12K+15K = 1

K = 1/36

2.

E(Y) = (1/36)*1+(3/36)*3+(5/36)*5+(12/36)*12+(15/36)*15 = 101/9 = 11.22

3.

Moment generating function ,

M(t) = summation of e^ty(y/36)=e^t*(1/36)+e^3t*(3/36)+e^5t*(5/36)+e^12t*(12/36)+e^15t*(15/36)

4.

V(Y) = M\'\'(0)-[M\'(0)]² = sum [ (t/36)e^ty(ty+2)-{(1/36)e^ty(ty+1)}²] put t=0

V(Y)= -5/36