Module 15: Moment Generating Functions

Question 1

what is the moment generating function?

Answer 1

The MGF of a random variable X is a function M_x (s) of a free parameter s defined by:

M_x (s) = E(e^(sx)) = SUM over all x (e^(sx) * p(x)) if discrete and integral from negative infinity to infinity e^(sx) f(x) dx if continuous

Question 2

what are the moment generating property?

Answer 2

If M(s) is the moment function of a random variable X, then

d^n / ds^n M(s) = M^(n)(0) = E(X^n)

For instance:
M(0) = 1; d/ds = M’(s) = E(X); M”(0) = E(X^2)

Question 3

what is the Linear Function property of MGF?

Answer 3

Let M(s) be the MGF of a random variable X. Then the MGF of the random variable Y = aX + b where a and b are constants, is: M_Y (s) = e^(sb) * M(as)

Question 4

what is the inversionn property/uniqueness property of MGFs?

Answer 4

The moment generating function M(s) of a random variable X completely determines it’s probability function. That is, if for all s

M_X (s) = M_Y (s)

then the random variables X and Y have the same probability function.