A) Given the probability function f(x).

Here are the solutions to your assignment:

a) Given the probability function f(x).
The given probability function is:

$$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}, \quad -\infty \le x \le \infty$$

This is the probability density function (PDF) of a Normal distribution with mean $\mu$ and variance $\sigma^2$.

• i) Find the moment generating function. The moment generating function (MGF) for a Normal distribution $X \sim N(\mu, \sigma^2)$ is: $$$$