The standard normal distribution

In a nutshell

The standard normal distribution is a normal distribution with mean $\mu = 0$ and variance $\sigma^2=1$ . Every other normal distribution can be transformed to the standard normal distribution using coding. It is important to know the standard normal distribution, $Z$ , as it can help solve problems that have unknown mean or standard deviation.

Notation

The standard normal distribution is often written as $Z$ . So $Z \sim (0,1)$ .

The cumulative probability $P(Z \le a)$ is often written as $\Phi (a)$ .

Coding with $Z$

If $X \sim N(\mu, \sigma^2)$ , then it is possible to transform each probability of $X$ into one in terms of $Z$ , where $Z \sim (0,1)$ using the formula:

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

Example 1

Given that $X\sim N(10,2.5^2)$ , and that $P(X \leq 7) = \Phi(a)$ , find the value of $a$ .

Rearrange the coding formula to make $X$ the subject:

$Z = \dfrac{X - \mu}{\sigma} \iff X = \mu + \sigma Z$

Substitute this into the probability involving $Z$ :

$\begin{aligned} P(X \le 7) &= P(\mu + \sigma Z \le 7)\\&= P(10 + 2.5Z \le 7)\\&= P(2.5Z \le -3)\\&=P\left(Z \leq -\dfrac{3}{2.5}\right)\\&=P(Z \le -1.2)\\&= \Phi (-1.2) \end{aligned}$ 

$a = \underline{-1.2}$ 

Note: The answer to $P(X \leq 7)$ under the normal distribution $X\sim N(10,2.5^2)$ should match the answer to $P(Z \le -1.2)$ or $\Phi (-1.2)$ under the standard normal distribution $Z \sim (0,1)$ which you can verify using a calculator as $0.115 \ (3 \ s.f.)$ .