Define Moment Generating Function.
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Definition Let X be a random variable. If the expected value [eq1] exists and is finite for all real numbers $t$ belonging to a closed interval [eq2], with $h>0$, then we say that X possesses a moment generating function and the function
[eq3]
is called the moment generating function of X.
The next example shows how the mgf of an exponential random variable is calculated.
Example Let X be a continuous random variable with support
[eq4]
and probability density function
[eq5]
where $lambda $ is a strictly positive number. The expected value [eq6] can be computed as follows:
[eq7]
Furthermore, the above expected value exists and is finite for any [eq8], provided [eq9]. As a consequence, X possesses a mgf:
[eq10]
[eq3]
is called the moment generating function of X.
The next example shows how the mgf of an exponential random variable is calculated.
Example Let X be a continuous random variable with support
[eq4]
and probability density function
[eq5]
where $lambda $ is a strictly positive number. The expected value [eq6] can be computed as follows:
[eq7]
Furthermore, the above expected value exists and is finite for any [eq8], provided [eq9]. As a consequence, X possesses a mgf:
[eq10]
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