TOPICS
Search

Moment-Generating Function


Given a random variable x and a probability density function P(x), if there exists an h>0 such that

 M(t)=<e^(tx)>
(1)

for |t|<h, where <y> denotes the expectation value of y, then M(t) is called the moment-generating function.

For a continuous distribution,

M(t)=int_(-infty)^inftye^(tx)P(x)dx
(2)
=int_(-infty)^infty(1+tx+1/(2!)t^2x^2+...)P(x)dx
(3)
=1+tm_1^'+1/(2!)t^2m_2^'+...,
(4)

where m_r^' is the rth raw moment.

For independent X and Y, the moment-generating function satisfies

M_(x+y)(t)=<e^(t(x+y))>
(5)
=<e^(tx)e^(ty)>
(6)
=<e^(tx)><e^(ty)>
(7)
=M_x(t)M_y(t).
(8)

If M(t) is differentiable at zero, then the nth moments about the origin are given by M^((n))(0)

M(t)=<e^(tx)>    M(0)=1
(9)
M^'(t)=<xe^(tx)>    M^'(0)=<x>
(10)
M^('')(t)=<x^2e^(tx)>    M^('')(0)=<x^2>
(11)
M^((n))(t)=<x^ne^(tx)>    M^((n))(0)=<x^n>.
(12)

The mean and variance are therefore

mu=<x>
(13)
=M^'(0)
(14)
sigma^2=<x^2>-<x>^2
(15)
=M^('')(0)-[M^'(0)]^2.
(16)

It is also true that

 mu_n=sum_(j=0)^n(n; j)(-1)^(n-j)mu_j^'(mu_1^')^(n-j),
(17)

where mu_0^'=1 and mu_j^' is the jth raw moment.

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

R(t)=ln[M(t)]
(18)
R^'(t)=(M^'(t))/(M(t))
(19)
R^('')(t)=(M(t)M^('')(t)-[M^'(t)]^2)/([M(t)]^2).
(20)

But M(0)=<1>=1, so

mu=M^'(0)=R^'(0)
(21)
sigma^2=M^('')(0)-[M^'(0)]^2=R^('')(0).
(22)

See also

Characteristic Function, Cumulant, Cumulant-Generating Function, Moment

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Moment-Generating Function

Cite this as:

Weisstein, Eric W. "Moment-Generating Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Moment-GeneratingFunction.html

Subject classifications