Question

What is moment generating function?

Answer 1

See below.

Explanation:

The parameters `mu` (mean) and `sigma` (standard deviation) locate the center and describe the spread associated with the values of a random variable `Y`. They do not, however, provide a unique characterization of the distribution of `Y`; many different distributions possess the same means and standard deviations.

• The `k`th moment of a random variable `Y` taken about the origin is defined to be `E(Y^k)` and is denoted by `mu_k^'`

• The moment-generating function `m(t)` for a random variable `Y` is defined to be `m(t)=E(e^(tY))`. We say that a moment-generating function for `Y` exists if there exists a positive constant `b` such that `m(t)` is finite for `abs(t) <= b`

If `m(t)` exists, then for any positive integer `k`,

`(d^km(t))/(dt^k)]_(t=0)=m^(k)(0)=mu_k^'`

In other words, if you find the `k`th derivative of `m(t)` with respect to `t` and then set `t=0`, the result will be `mu_k^'`.

Then we find that various probability distributions have their own unique moment-generating function.