Question

Let X, Y, and Z be independent and identically distributed discrete random variables, with each having a probability distribution that puts a mass of 1/4 on the number 0, a mass of 1/4 at 1, and a mass of 1/2 at 2.

a. Compute the moment generating function for S= X+Y+Z

b. Use the MGF from part a to compute the second moment of S, E(S^2)

c. Compute the second moment of S in a completely different way, by expanding the square in the expression (X+Y+Z)^2 and applying the expectation operator.

Answer 1

The moment generating function of X is:

Like wise:

a)

Since X, Y and Z are independent so the MGF of S is

b)

Differentiating above with respect to t gives

Differentiating above with respect to t again gives

Putting t=0 gives

Answer: 16.125

c)

FIrst we need to find expectation of X and expectation X^{2}. SO,

X	P(X=x)	xP(X=x)	x^2P(X=x)
0	0.25	0	0
1	0.25	0.25	0.25
2	0.5	1	2
Total		1.25	2.25

So,

Likwise:

Now: