2 The standard normal distribution

At this stage we shall, for simplicity, consider what is known as a standard normal distribution which is obtained by choosing particularly simple values for $\mu$ and $\sigma$ .

In Figure 2 we show the graph of the standard normal distribution which has probability density function $y=\frac{1}{\sqrt{2\pi }}{e}^{-x^2∕2}$

Figure 2 :

The result which makes the standard normal distribution so important is as follows:

Example 1

If the random variable $X$ is described by the distribution $N\left(45,0.000625\right)$ then what is the transformation required to obtain the standardised normal variable?

Solution

Here, $\mu =45$ and ${\sigma }^2=0.000625$ so that $\sigma =0.025$ . Hence $Z=\left(X-45\right)∕0.025$ is the required transformation.

Example 2

When the random variable $X\sim N\left(45,0.000625\right)$ takes values between 44.95 and 45.05, between which values does the random variable $Z$ lie?

Solution

When $X=45.05,$ $Z=\frac{45.05-45}{0.025}=2$

When $X=44.95,$ $Z=\frac{44.95-45}{0.025}=-2$

Hence $Z$ lies between $-2$ and 2.

Task!

The random variable $X$ follows a normal distribution with mean 1000 and variance 100. When $X$ takes values between 1005 and 1010, between which values does the standardised normal variable $Z$ lie?

Answer