4.6: Normal Distribution

Definition$$\PageIndex{1}$$

A random variable $$X$$ has a normal distribution, with parameters $$\mu$$ and $$\sigma$$, write $$X\sim\text{normal}(\mu,\sigma)$$, if it has pdf given by
$$f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}, \quad\text{for}\ x\in\mathbb{R},\notag$$
where $$\mu\in\mathbb{R}$$ and $$\sigma > 0$$.

The normal distribution is arguably the most important probably distribution. It is used to model the distribution of population characteristics such as weight, height, and IQ. The pdf is terribly tricky to work with, in fact integrals involving the normal pdf cannot be solved exactly, but rather require numerical methods to approximate. Because of this, there is no closed form for the corresponding cdf of a normal distribution. Given the importance of the normal distribution though, many software programs have built in normal probability calculators. There are also many useful properties of the normal distribution that make it easy to work with. We state these properties without proof below.

Properties of the Normal Distribution

1. If $$X\sim\text{normal}(\mu, \sigma)$$, then $$aX+b$$ also follows a normal distribution with parameters $$a\mu + b$$ and $$a\sigma$$.
2. If $$X\sim\text{normal}(\mu, \sigma)$$, then $$\displaystyle{\frac{X-\mu}{\sigma}}$$ follows the standard normal distribution, i.e., the normal distribution with parameters $$\mu=0$$ and $$\sigma = 1$$.​​​​​​​​​​​​​​