Moment Generating Function/Part1

Moment Generating Function
It is a function which is defined as follows: M(t)=E(exp(tx)) = sum of the prdduct of exp(tx)by the probability mass functio of adiscrete random variable over all values of this random variable,
and
it = the integral of the producd of exp(tx) by the probability density function of the continuous random variable.
sum of the prdduct of exp(tx) of n random variables= the producte of the moment generating functions The moment generating function of sum of n random variables =the product of the moment generating functions of these random variables.
The moment generating function of the product of constant c by thr random variable or any function of this variable = the product of this constant by exp(tx)
We can know the distributioin of such random variable according to its moment generating function
The moment generating function is aspecial case of the characteristic function,it does not exist for some variables
The characteristic function =E(exp(itx))
The logarithm of the moment generating function is called the cumulant- generating function