The Normal Distribution

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.