2.Moment  generating  function  for  discrete  random variable 

The moment generating function M(t) of the discrete random variable X is defined for all real values of t by 

If X is discrete with mass function 𝑝(𝑥)We  call  M(t)  the  moment  generating  function  because

all  of  the moments of X can be obtained by successively differentiating M(t) and then evaluating the result at t = 0. For example, 

where we have assumed that the interchange of the differentiation and

expectation operators is legitimate. That is, we have assumed that

This assumption can almost always be justified. 

Similsarly,

Thus, 

In general, the nth derivative of M(t) is given by

implying that 

We now compute M(t) for some common discrete distributions. 

Binomial distribution with parameters n and pIf X is a binomial random variable with parameters n and p, then 

Differentiating a second time yields 

The variance of X is given by 

Poisson distribution with mean λIf X is a Poisson random variable with parameter λ, then 

Differentiation yields 

Hence, both the mean and the variance of the Poisson random variable equal λ.

The following table give the moment generating functions for some common discrete distributions.

An  important  property  of  moment  generating  functions  is  that  the moment

generating  function  of  the  sum  of  independent  random variables  equals  the  product

of  the  individual  moment  generating functions. To prove this,

suppose that X and Y areindependent and have moment  generating  functions

 respectively.  Then  the moment generating function of X + Y, is given by

Another  important  result  is  that  the  moment  generating  function uniquely  determines  the  distribution.

That  is,  if   exists  and  is finite in some region about t = 0,

then the distribution of X is uniquely determined. For instance, if 

then it follows from the last Table that X is a binomial random variable with parameters 10 and 

Suppose that the moment generating function of a random variable X is given by

We can see from last Table that is the moment generating function of a Poisson

random variable with mean 3. Hence, by the one-to-one correspondence between moment generating functions

and distribution functions, it follows that X must be a Poisson random variable with mean 3.

Thus, 

Sums of independent binomial random variablesIf X and Y are

independent binomial random variables with parameters (n, p) and (m, p), respectively, what is the distribution of X + Y ?

The moment generating function of X + Y is given by

However, is the moment generating function of a binomial random variabl

e having parameters m + n and p. Thus, this must be the distribution of X + Y.

Sums of independent Poisson random variablesCalculate the distribution of X + Y when X and Y are

independent Poisson random variables with means respective