Moment-generating Function - Definition

Definition

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

wherever this expectation exists.

always exists and is equal to 1.

A key problem with moment-generating functions is that moments and the moment-generating function may not exist, as the integrals need not converge absolutely. By contrast, the characteristic function always exists (because it is the integral of a bounded function on a space of finite measure), and thus may be used instead.

More generally, where T, an n-dimensional random vector, one uses instead of tX:

The reason for defining this function is that it can be used to find all the moments of the distribution. The series expansion of etX is:

e^{tX} = 1 + tX + \frac{t^2X^2}{2!} + \frac{t^3X^3}{3!} + \cdots +\frac{t^nX^n}{n!} + \cdots.

Hence:

M_X(t) = E(e^{tX}) = 1 + tm_1 + \frac{t^2m_2}{2!} + \frac{t^3m_3}{3!}+\cdots + \frac{t^nm_n}{n!}+\cdots,

where mn is the nth moment.

If we differentiate MX(t) i times with respect to t and then set t = 0 we shall therefore obtain the ith moment about the origin, mi.

Read more about this topic:  Moment-generating Function

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