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Encyclopedia > Moment generating function

In probability theory and statistics, the moment-generating function of a random variable X is

M_X(t)=E\left(e^{tX}\right).

The moment-generating function generates the moments of the probability distribution, as follows:

E\left(X^n\right)=M_X^{(n)}(0)=\left.\frac{\mathrm{d}^n}{\mathrm{d}t^n}\right|_{t=0} M_X(t).

If X has a continuous probability density function f(x) then the moment generating function is given by

M_X(t) = \int_{-\infty}^\infty e^{tx} f(x)\,\mathrm{d}x
= \int_{-\infty}^\infty \left( 1+ tx + \frac{t^2x^2}{2!} + \cdots\right) f(x)\,\mathrm{d}x
= 1 + tm_1 + \frac{t^2m_2}{2!} +\cdots,

where mi is the ith moment.


Regardless of whether probability distribution is continuous or not, the moment-generating function is given by the Riemann-Stieltjes integral

\int_{-\infty}^\infty e^{tx}\,dF(x)

where F is the cumulative distribution function.


If X1, X2, ..., Xn is a sequence of independent (and not necessarily identically distributed) random variables, and

S_n = \sum_{i=1}^n a_i X_i,

where the a i are constants, then the probability density function for S n is the convolution of the probability density functions of each of the X i and the moment-generating function for S n is given by

M_{S_n}(t)=M_{X_1}(a_1t)M_{X_2}(a_2t)\ldots M_{X_n}(a_nt).


Related concepts include the characteristic function, the probability-generating function, and the cumulant-generating function. The cumulant-generating function is the logarithm of the moment-generating function.
 

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