Thread: Moment Generating Function of geometric rv

What is the mgf of the geometric random variable with pmf of f(w) = p(1-p)^(w-1), where p = 1/8? Show how to derive it and explain any bounds on the values of t for which it is valid.


So far, I have gotten it simplified to:
M(t) = (1/7) * Sigma( ((7/8)*e^t)^w ) where the sum is from w=0 to infinity.
I don't really know where to go from here to simplify it, or what bounds t could possibly have.

Originally Posted by uberbandgeek6

What is the mgf of the geometric random variable with pmf of f(w) = p(1-p)^(w-1), where p = 1/8? Show how to derive it and explain any bounds on the values of t for which it is valid.


So far, I have gotten it simplified to:
M(t) = (1/7) * Sigma( ((7/8)*e^t)^w ) where the sum is from w=0 to infinity.
I don't really know where to go from here to simplify it, or what bounds t could possibly have.

In the form of the pmf you have given, x = 1, 2, 3, ..., so the sum should run from 1 to infinity.

It looks like a geometric series to me. Do you know the formula for the sum of a geometric series?

It's been a while since I used that. So then it would be 1/(7 - (49/8)e^t) where t != ln(8/7) ?

Not quite. The simplest formula for the sum of a geometric series is



but your series doesn't start with 1, so you will need to factor out to get it into the appropriate form.

Okay, so the sum of a geometric sequence is a/(1-r) where a is the first term and r is the ratio, correct? Wouldn't the first term be when w=0, so ((7/8)e^t)^0 = 1 (ignoring for the moment that I moved the 1/7 outside the sum, of course)? And since ((7/8)e^t) is to the power of w, wouldn't the ratio be (7/8)e^t?

Originally Posted by uberbandgeek6

Okay, so the sum of a geometric sequence is a/(1-r) where a is the first term and r is the ratio, correct? Wouldn't the first term be when w=0, so ((7/8)e^t)^0 = 1 (ignoring for the moment that I moved the 1/7 outside the sum, of course)? And since ((7/8)e^t) is to the power of w, wouldn't the ratio be (7/8)e^t?

No, in the form you have given for the distribution function, the first term is when w=1.