Then:\(M(t)=E(e^{tX})=\sum\limits_{x\in S} e^{tx}f(x)\)is the moment generating function of \(X\) as long as the summation is finite for some interval of \(t\) around 0.
Also note that equality of the distribution functions can be replaced in the
proposition above by:
equality of the probability mass functions (if
and
are discrete random
variables);
review
equality of the probability density functions (if
and
are continuous
random variables). g. Online appendix. That is:Before we prove the above proposition, recall that \(E(X), E(X^2), \ldots, E(X^r)\) are called moments about the origin.
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Here are some examples of the moment-generating function and the characteristic function for comparison. . . Besides helping to find moments, the moment generating function has an important property often called the uniqueness property.
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Denote by
and
their distribution
functions and by
and
their mgfs.
Let
X
{\displaystyle X}
be a random variable with CDF
F
X
{\displaystyle F_{X}}
.
The moment-generating function is so called because if it exists on an open interval around t=0, then it is the exponential like this function of the moments of the probability distribution:
That is, with n being a nonnegative integer, the nth moment about 0 is the nth derivative of the moment generating function, evaluated at t = 0. .
Not all random variables possess a moment generating function.
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The most important property of the mgf is the following.
Derive the moment generating function of
,
if it exists. f. We use \(\hat{p}\) to denote the estimate of \(p\). This is a preview of subscription content, access via your institution.
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By the definition of moment generating
function, we
haveObviously,
the moment generating function exists and it is well-defined because the above
expected value exists for any
. .