Question

Let the probability function of X be given by f(x)= ce^(-x) , x=1,2,3, ...
How to find the moment generating function of X and its E(X)?

Answer 1

Let X = random variable, t = a dummy variable, f(x)=pdf of discrete random variable X

Moment generating function, \(M_X(t)\)
\(=E[e^{tX}]\) (expected value of e^(tX))
\(=\sum_{-\infty}^\infty e^{tx}f(x)dx\)
(note: limits of summation cover the actual domain of f(x), in this case, sum from 1 to \(\infty\) )

E(X) is just \(M_X(0)\) once you have found \(M_X(t)\).

For more theoretical background, read for example:
https://en.wikipedia.org/wiki/Moment-generating_function