Given a Random Variable
, if there exists an
such that
![\begin{displaymath}
M(t) \equiv \left\langle{e^{tx}}\right\rangle{} = \cases{ \s...
...fty}^\infty e^{tx}P(x)\,dx & for a continuous distribution\cr}
\end{displaymath}](m_1716.gif) |
(1) |
for
, then
![\begin{displaymath}
M(t) \equiv \langle e^{tx}\rangle
\end{displaymath}](m_1718.gif) |
(2) |
is the moment-generating function.
where
is the
th Moment about zero. The moment-generating function satisfies
If
is differentiable at zero, then the
th Moments about the Origin are given by
![\begin{displaymath}
M(t) = \langle e^{tx}\rangle \qquad M(0) = 1
\end{displaymath}](m_1728.gif) |
(5) |
![\begin{displaymath}
M'(t) = \langle xe^{tx}\rangle \qquad M'(0) = \langle x\rangle
\end{displaymath}](m_1729.gif) |
(6) |
![\begin{displaymath}
M''(t) = \langle x^2e^{tx}\rangle \qquad M''(0) = \langle x^2\rangle
\end{displaymath}](m_1730.gif) |
(7) |
![\begin{displaymath}
M^{(n)}(t) = \langle x^ne^{tx}\rangle \qquad M^{(n)}(0)=\langle x^n\rangle.
\end{displaymath}](m_1731.gif) |
(8) |
The Mean and Variance are therefore
It is also true that
![\begin{displaymath}
\mu_n=\sum_{j=0}^n{n\choose j}(-1)^{n-j}\mu_j'(\mu'_1)^{n-j},
\end{displaymath}](m_1734.gif) |
(11) |
where
and
is the
th moment about the origin.
It is sometimes simpler to work with the Logarithm of the moment-generating function, which is also called the
Cumulant-Generating Function, and is defined by
But
, so
See also Characteristic Function, Cumulant, Cumulant-Generating Function, Moment
References
Kenney, J. F. and Keeping, E. S. ``Moment-Generating and Characteristic Functions,'' ``Some Examples of Moment-Generating Functions,''
and ``Uniqueness Theorem for Characteristic Functions.'' §4.6-4.8 in Mathematics of Statistics, Pt. 2, 2nd ed.
Princeton, NJ: Van Nostrand, pp. 72-77, 1951.
© 1996-9 Eric W. Weisstein
1999-05-26