5-  Moment  generating  functions  for  continuous  random variable 

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

If X is continuous with density 𝑓(𝑥).

As  in  discrete  case,  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  

in the continuous case. This assumption can almost always be justified.

As in discrete case, from the equation  evaluated at 𝑡 =0, we obtain  

Similsarly,

Thus,

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

implying that

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

Exponential distribution with parameter λ

We note from this derivation that, for the exponential distribution,

M(t)is defined only for values of t less than λ. Differentiation of M(t)yields

Hence, 

The variance of X is given by 

Normal distributionWe first compute the moment generating function of a unit normal

random variable with parameters 0 and 1. Letting Z be such a random variable, we have

Hence,  the  moment  generating  function  of  the  unit  normal 

random variable Z is given by 

To obtain the moment generating function of an arbitrary normal random variable,

we recall that   X = μ + σ Z will have a normal distribution

with parameters μ and  whenever Z is  a  unit  normal  random  variable.

Hence,  the  moment generating function of such a random variable is given by 

By differentiating, we obtain 

Thus,

implying that 

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

As in discrete case 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. 

Sums of independent normal random variablesShow that if X and Y are

independent normal random variables with respective parameters  and

then X + Y is normal with mean  and variance 

which is the moment generating function of a normal random variable with

mean  and variance 

The desired result then follows because the moment generating function uniquely determines the distribution.