Normal Random Variable

If we take a an affine transformation of a standard Normal random variable: Y=aZ+b the new density of Y is

$$f(y)=\frac{1}{\sqrt{2 \pi a^2 }} e^{-\frac{1}{2}(\frac{y-b}{a})^2}$$

This is called the Normal variable with parameters, b and a², denoted by ${\cal N}(b,a^2)$.

For any such transformation we have: if $Y=\sigma Z + \mu$, then the density of Y is

$$f(y)=\frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{(y-\mu)^2}{2 \sigma^2}}$$

The two parameters that are needed to define a normal are: $\mu=E[Z]$, $\sigma=var(Z)$, this explanation will be developed in chapter 6.

In general if you have a Normal random variable with parameters $\mu$ and $\sigma^2$, we need to standardize it, because the probabilities cannot be computed from a closed form formula, this is done by standardizing, say $Y \sim {\cal N}(b,a^2)$

$$P(C \leq Y \leq D)=P( \frac{C-b}{a} \leq \frac{Y-b}{a} \leq \frac{D-b}{a} )$$

Now $Z=\frac{Y-b}{a}$ is a standard ${\cal N}(0,1)$ variable so we can use the distribution function $\Phi$

$$P(C \leq Y \leq D)=\Phi( \frac{D-b}{a})- \Phi( \frac{C-b}{a})$$

A web site that allows you to look up some probabilities for Normal distributions:

Probabilities for Normal