Introduction

The Normal distribution is the most important of all distributions that are used in statistics. In many cases it serves as a generic model for the distribution of a measurement. Moreover, even in cases where the measurement is modeled by other distributions (i.e. Binomial, Poisson, Uniform, Exponential, etc.) the Normal distribution emerges as an approximation of the distribution of numerical characteristics of the data produced by such measurements.

The PDF

The notation \(X \sim N(\mu,\sigma)\) means that \(X\) has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). The normal density function is a bell shaped distribution that is symmetric around its mean \(\mu\).

The formula for the probability density function for the normal distribution is

\[f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{2} \left( \frac{x-\mu}{\sigma}\right)^2} \text{ for } -\infty \le x \le \infty\]

The CDF

\[F(x_0) = P(X \le x_0) = \int_{-\infty}^{x_0} f(x) dx = \int_{-\infty}^{x_0} \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{2} \left( \frac{x-\mu}{\sigma}\right)^2} dx\]

This is not an easy integral to evaluate so we will only use R to compute normal probabilities with the command \(P(X \le x) = \text{pnorm}(x,\mu,\sigma)\)

The Standard Normal Distribution

A special case of the normal distribution is the standard normal distribution which is the normal distribution with mean \(\mu = 0\) and standard deviation \(\sigma = 1\) and is denoted by the letter \(Z\). So we write \[Z \sim N(0,1)\]

Any normal random variable can be rescaled into a standard normal random variable and this sometimes is useful in solving certain problems. If \(X \sim N(\mu,\sigma)\) then

\[Z = \frac{X-\mu}{\sigma} \sim N(0,1)\]

Example 1:

Suppose \(X \sim N(5,2)\). Find

  • \(P(X < 3.4)\)
  • \(P(X > 8.6)\)
  • \(P(3 < X < 7)\)

  • \(P(X < 3.4) = \text{pnorm}(3.4,5,2) = 0.212\) or in terms of \(Z\) we have \(P(X < 3.4) = P(Z < \frac{3.4-5}{2}) = P(Z<-0.8)=\text{pnorm}(-0.8) = 0.212\)
  • \(P(X > 8.6) = 1 - P(X < 8.6) = 1 - \text{pnorm}(8.6,5,2) = 0.036\) or in terms of \(Z\) we have \(P(X > 8.6) = P(Z > \frac{8.6-5}{2}) = P(Z>1.8)=1-\text{pnorm}(1.8) = 0.036\)
  • \(P(3 < X < 7) = P(X<7)-P(X<3) = \text{pnorm}(7,5,2)-\text{pnorm}(3,5,2) = 0.683\)

Finding Percentiles

The \(p^{th}\) percentile of a distribution is the point at which \(p%\) of the observations lie below. For example, your score on a standardized exam is often expressed as a percentile indicating what percent of all students scored lower than your score.

If \(X \sim N(\mu,\sigma)\) then the \(p^{th}\) percentile of \(X\) can be found using R with the qnorm command: \(\text{qnorm}(p,\mu,\sigma)\)

Example 2:

Suppose \(X \sim N(2,0.1)\). Find the \(90^{th}\) percentile of \(X\)

qnorm(0.9,2,0.1) = 2.128155