The Normal Distribution

Students learn to compute probabilities for a normally distributed random variable $X \sim N(\mu, \sigma^2)$ by standardizing to the $Z$ -score and using values of the standard normal cumulative distribution function $\Phi$ . The lesson covers left-tail, right-tail, and between-values probabilities.

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Tutorial

The Normal Distribution and Standardization

A continuous random variable $X$ follows a normal distribution with mean $\mu$ and variance $\sigma^2$ , written

X \sim N(\mu, \sigma^2),

if its density is the bell-shaped curve

f(x) = \dfrac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^2}{2\sigma^2}}.

The curve is symmetric about $\mu$ , and the standard deviation $\sigma$ controls its spread.

To compute a probability like $P(X \le a)$ , we standardize by converting $X$ to its $Z$ -score:

Z = \dfrac{X - \mu}{\sigma} \;\sim\; N(0,1).

This lets us rewrite any normal probability in terms of the standard normal CDF $\Phi$ :

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

For example, if $X \sim N(10, 4)$ , then $\sigma = \sqrt{4} = 2$ , and

P(X \le 13) = \Phi\!\left(\dfrac{13-10}{2}\right) = \Phi(1.5) = 0.9332.

Notice that we always divide by $\sigma$ , not $\sigma^2$ .