Modeling With the Normal Distribution

A continuous random variable $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$ if its values cluster symmetrically around $\mu$ with spread controlled by $\sigma$. We write $X \sim N(\mu, \sigma^2)$.

Many real-world quantities -- heights, exam scores, measurement errors, manufacturing tolerances -- are well approximated by a normal model. To compute probabilities for $X$, we standardize:

$$Z = \dfrac{X - \mu}{\sigma}.$$

The new variable $Z$ follows the standard normal distribution $N(0, 1)$, whose cumulative distribution function we denote by $\Phi$. Tables (or calculators) give $\Phi(z) = P(Z < z)$ for any $z$.

To find $P(X < a)$, we standardize the cutoff:

$$P(X < a) = P\!\left(Z < \dfrac{a - \mu}{\sigma}\right) = \Phi\!\left(\dfrac{a - \mu}{\sigma}\right).$$

For example, if $X \sim N(50, 25)$, then $\sigma = 5$, and

$$P(X < 55) = \Phi\!\left(\dfrac{55 - 50}{5}\right) = \Phi(1) = 0.8413.$$