Question

Suppose that Z is a continuous random variable. Let $\phi (z)$ denote the unnormalized PDF of Z ―the function $\phi$ satisfies all properties of a PDF except that it is not normalized. Now suppose we use $\phi$ to compute something like the moment generating function (MGF), i.e., we compute the function

$\gamma (s)= \int_{-\infty}^{\infty} e^{sz}\phi(z)dz$

What is $\gamma (0)$? How can we use $\gamma (0)$ to normalize the PDF?

Answer 1

let constant = k

such that

$\int_{-\infty }^{\infty }k \phi(z)dz = 1$

$\Rightarrow k \int_{-\infty }^{\infty } \phi(z)dz = 1$

$\Rightarrow k \gamma (0)= 1$

$\Rightarrow k = \frac{1}{\gamma (0)}$

hence

we have divide $\phi (z)$ by $\gamma (0)$   to normalize the pdf