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# What is moment generating function?

Dec 31, 2017

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 $E \left({Y}^{k}\right)$ and is denoted by ${\mu}_{k}^{'}$

• The moment-generating function $m \left(t\right)$ for a random variable $Y$ is defined to be $m \left(t\right) = E \left({e}^{t Y}\right)$. We say that a moment-generating function for $Y$ exists if there exists a positive constant $b$ such that $m \left(t\right)$ is finite for $\left\mid t \right\mid \le b$

If $m \left(t\right)$ 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 $m \left(t\right)$ 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.