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 $$0.115 \ (3 \ s.f.)$$.

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