###### 3.10.1 Normal Distributions

A **normal distribution** is specified by two parameters: a mean μ and variance σ^{2}. We denote it *N*(μ,σ^{2}). Its PDF is

[3.91]

This is graphed in Exhibit 3.15:

Irrespective of its mean or standard deviation, every normal distribution has skewness and kurtosis

[3.92]

[3.93]

With a kurtosis of 3, normal distributions fall precisely between platykurtosis and leptokurtosis. Distributions that have lower kurtosis than a normal distribution are platykurtic. Those that have higher kurtosis are leptokurtic.

A linear polynomial of a normal random variable is also normal. If *X* ~ *N*(μ,σ^{2}),

[3.94]

for any constants *a*, *b* ∈ . This means that any *N*(μ,σ^{2}) random variable *X* can be expressed as a linear polynomial of some *N*(0,1) random variable* Z*:

[3.95]

We call *N*(0,1) the **standard normal distribution**.

It has been proven that there is no closed-form expression for the CDF Φ of a normal distribution. The function exists. It simply cannot be expressed in terms of other standard functions. In practice, it and its inverse Φ^{–1} are approximated to many decimal places using computer algorithms. See Patel (1996).

Based upon [3.95], it follows that any quantile of an *N*(μ,σ^{2}) distribution occurs a distance from its mean μ that is a fixed multiple of σ. For example, the .90-quantile of a standard normal variable *Z* is obtained from a standard normal table as 1.282. Then, for any *N*(μ,σ^{2}) random variable *X*, the .90-quantile occurs 1.282 standard deviations σ above its mean μ because, by [3.95],

[3.96]

[3.97]

[3.98]

The result is independent of the values of μ and α. Accordingly, the .90-quantile of any *N*(μ,σ^{2}) random variable is 1.282 standard deviations σ greater than its mean μ. Results for other quantiles are shown in Exhibit 3.16:

Because a normal distribution is symmetrical about its mean, the .10, .05, .025, and .01 quantiles can be obtained by replacing plus signs with minus signs in Exhibit 3.16.

*N*(μ,σ

^{2}) distribution.

For large *m*, the binomial and normal distributions approximate one another as follows:

[3.99]