# Moment generating function and expectation

## Homework Statement

Let X denote a random variable with the following probability mass function:
P(j)= 2^(-j), j=1,2,3,.....
(a) Compute the moment generating function of X.
(b) Use your answer to part (a) to compute the expectation of X.

## Homework Equations

m.g.f of X is M (t) = E[e^tX]

## The Attempt at a Solution

I computed the MGF (X) to be: e^t/(2-e^t). I need a suggestion about the next step. I'm confused about the relationship between the MGF and expectation of X.

Thanks!

tiny-tim
Homework Helper
Hi BookMark440! Does this help … http://en.wikipedia.org/wiki/Moment_generating_function ? That link was very helpful. I completed the problem but I am a little uncertain about my strategy of finding the derivative by parts and the answer:

X has p.d.f : p(j) = 2^(-j), j=1,2,3,....

I computed the MGF for X = e^t/(2-e^t)

Then E[X] = d/dt (e^t/(2-e^t)) [evaluated for t= 0] =

numerator = (2-e^t)d/dt(e^t) - (e^t)d/dt(2-e^t)
denominator = (2-e^t)^2

Evaluating for t=0, the final answer is:
E[X] = 2

Is this the correct strategy and answer?

Thanks!

tiny-tim
Homework Helper
Hi BookMark440! (try using the X2 tag just above the Reply box )
Then E[X] = d/dt (e^t/(2-e^t)) [evaluated for t= 0] =

numerator = (2-e^t)d/dt(e^t) - (e^t)d/dt(2-e^t)
denominator = (2-e^t)^2

Evaluating for t=0, the final answer is:
E[X] = 2

Is this the correct strategy and answer?!

Looks good! 