Question

Write short note on moment generating function.

Answer 1

The moment - generating function of a random variable X . Where it exists is given by Mₓ(t) = E[ exp(tX)] or E(e^tX) = ∑e^tX. f(x) when x is continuous .
And Mₓ(t) = $E(e^{tX})=\int\limits^{\infty}_{-\infty}{e^{tX}f(x)}\,dx$ when x is discrete random variable .

To explain why we prefer to this function as an moment - generating function.Lets substitute for $e^{tX}$ its Maclaurin's series expansion.
e.g., $e^{tX}= 1 + tX + \frac{t^2X^2}{2!}+\frac{t^3X^3}{3!}+......\frac{t^rX^r}{r!}+....$
For discrete case , we get
Mₓ(t) = $\sum{[1 + tX + \frac{t^2X^2}{2!}+\frac{t^3X^3}{3!}+......\frac{t^rX^r}{r!}+....]}f(x)$
= $1 + \mu t + \mu_2' \frac{t^2}{2!} + ...... + \mu_r'\frac{t^r}{r!}+.....$

It can be seen that Maclaurin's series of the moment generating function of X , the coefficient of $\frac{t^r}{r!}$ is $\mu_r'$ , the rth moment about the origin . You can check For continuous case , we will get argument is the same for both cases.