####
__Definition__

Let
be a random variable. If the
expected value
exists and is finite for all real numbers
belonging to a closed interval
,
with
,
then we say that
possesses a moment generating function and the
functionis
called the

####

Let

**moment generating function**of####
__Definition__

Let *X*be a random variable for an experiment taking values in a subset*S*of**R.**The moment generating function of*X*is the function*M*_{X}defined byM_{X}(t) =E[exp(tX)] fortinR

####
__Properties__

In the following exercises, assume that the moment generating
functions are finite in an interval about 0.**3.**Show that for any nonnegative integer

*n*,

Thus, the derivatives of the moment generating function at 0 determine the moments of the variable (hence the name).M_{X}^{(n)}(0) =E(X^{n})

**4.**If

*a*and

*b*are constants, show that

M_{aX + b}(t) = exp(bt)M_{X}(at)

**5.**Suppose that

*X*and

*Y*are independent. Show that

For more, click here.M_{X + Y}(t) =M_{X}(t)M_{Y}(t)

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