The MomentGeneratingFunction function computes the moment generating function of the specified random variable at the specified point.

The first parameter can be a random variable or an algebraic expression involving random variables (see Student[Statistics][RandomVariable]).

By default, all computations involving random variables are performed symbolically (see option numeric below).

If the input point is a floating point value, then the output will be given in a floating point. Otherwise, the output is exact.

The value of the moment generating function of the input random variable at the specified point is computed according to the rules mentioned above. To always compute the outcome numerically, specify the numeric or numeric=true option.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Statistics}\right]\right)\:$

$\mathrm{MomentGeneratingFunction}\left(\mathrm{BinomialRandomVariable}\left(n\,p\right)\,t\right)$

${\left(p{\ⅇ}^t+1-p\right)}^n$

$\mathrm{MomentGeneratingFunction}\left(\mathrm{BinomialRandomVariable}\left(10\,\frac{1}{2}\right)\,3\right)$

${\left(\frac{{\ⅇ}^3}{2}+\frac{1}{2}\right)}^{10}$

$\mathrm{MomentGeneratingFunction}\left(\mathrm{BinomialRandomVariable}\left(10\,0.5\right)\,3\,\mathrm{numeric}\right)$

$1.696471963\times {10}^{10}$

$\mathrm{MomentGeneratingFunction}\left(\mathrm{BinomialRandomVariable}\left(10\,0.5\right)\,3.0\right)$

$1.696471963\times {10}^{10}$

$\mathrm{MomentGeneratingFunction}\left(\mathrm{BinomialRandomVariable}\left(10\,\frac{1}{2}\right)\,3\,\mathrm{inert}\right)$

$\sum _{\mathrm{\_t}=0}^{10}{\ⅇ}^{3\mathrm{\_t}}\left(\genfrac{}{}{0ex}{}{10}{\mathrm{\_t}}\right){\left(\frac{1}{2}\right)}^{\mathrm{\_t}}{\left(\frac{1}{2}\right)}^{10-\mathrm{\_t}}$

$\mathrm{evalf}\left(\mathrm{MomentGeneratingFunction}\left(\mathrm{BinomialRandomVariable}\left(10\,\frac{1}{2}\right)\,3\,\mathrm{inert}\right)\right)$

$1.696471965\times {10}^{10}$

$X≔3\mathrm{PoissonRandomVariable}\left(a\right)+2\mathrm{ExponentialRandomVariable}\left(b\right)\:$

$\mathrm{MGF}\left(X\,t\right)$

$\frac{{\ⅇ}^{a\left({\ⅇ}^{3t}-1\right)}}{-2bt+1}$

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

The Student[Statistics][MomentGeneratingFunction] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.