# The Normal Distribution

Posted by Beetle B. on Tue 06 June 2017

## Probability Distribution Function

The parameters are $$-\infty<\mu<\infty$$ and $$\sigma>0$$.

\begin{equation*} f\left(x;\mu,\sigma\right)=\frac{1}{\sqrt{2\pi n}\sigma}\exp\left(-\frac{\left(x-\mu\right)^{2}}{2\sigma^{2}}\right) \end{equation*}

It is often denoted as $$X\sim N\left(\mu,\sigma^{2}\right)$$

## Mean and Variance

The mean is $$\mu$$ and variance is $$\sigma^{2}$$.

## Standard Normal Distribution

When $$\mu=0,\sigma=1$$, then this is called the standard normal distribution and $$X$$ is the standard normal random variable, usually denoted by $$Z$$.

The cdf is denoted by $$\Phi(z)$$

Given a normal distribution, we can transform it into the standard normal distribution using:

\begin{equation*} Z=\frac{X-\mu}{\sigma} \end{equation*}

### Critical Values

$$z_{\alpha}$$ is used to denote the value of $$Z$$ for which the area under the curve to the right is $$\alpha$$. Effectively, it is the value $$z$$ such that $$P(Z\ge z_{\alpha})=\alpha$$. They are referred to critical values.

## Some Properties

• 68% of the values are within 1 $$\sigma$$ of the mean.
• 95% of the values are within 2 $$\sigma$$ of the mean.
• 99.7% of the values are within 3 $$\sigma$$ of the mean.

## Approximation For a Discrete Distribution

We often use the normal distribution to approximate a discrete one. But exercise caution! Say you want the probability that the IQ is greater than 125. Note that the IQ is an integer. Don’t compute $$P(X\ge125)$$. Instead, compute $$P(X\ge124.5)$$

This is called a continuity correction.

### Binomial Approximation

We often approximate binomial distributions with normal ones. But do note: The Binomial distribution is skewed for $$p\ne 0.5$$, but the normal distribution is never skewed. We use the same mean and standard deviation as the Binomial one. The approximation is good enough when both $$np\ge10$$ and $$nq\ge10$$

\begin{equation*} P\left(X\le x\right)=B(x;n,p)=\Phi\left(\frac{x+0.5-np}{\sqrt{npq}}\right) \end{equation*}

## Linear Transformation

If we transform the normal distribution with $$Y=aX+b$$, then the distribution for $$Y$$ is also normal.