Moment Generating Function

Moment generating function
The distribution of a random variable is often characterized in terms of its moment generating function (mgf), a real function whose derivatives at zero are equal to the moments of the random variable.
Moment generating functions have great practical relevance not only because they can be used to easily derive moments, but also because a probability distribution is uniquely determined by its mgf, a fact that, makes them a handy tool to solve several problems, such as deriving the distribution of a sum of two or more random variables.

It must be mentioned that not all random variables possess a moment generating function. However, all random variables possess a characteristic function.
Def: Let X be a discrete random variable. Then the moment generating function of X is the function of the variable t defined as
mX(t) = E(etX)
• the moment generating function is the expected value of the function etX
• the variable t is just a parameter (auxiliary variable), whose use will become clear
• the moments of X are hidden inside the function mX(t)!