The standard normal distribution

If the normal distribution has \(\mu=0\) and \(\sigma^2=1\), it is called a standard normal distribution and is denoted as \(Z \sim N(0,1)\). The probability density function (pdf) for the standard normal distribution is given by

\[\mathrm{f}(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}\]

In general, any normal distribution \(X \sim N(\mu,\sigma^2)\) can be transformed into the standard normal \(Z \sim N(0,1)\) by using the transformation

\[Z = \frac{X-\mu}{\sigma}\]

For example, if \(X \sim N(\mu, \sigma^2)\) and we want to find \(P(x_1 \leq X \leq x_2)\), then \(X\) must first be transformed to \(Z \sim N(0, 1)\). Then \(P(x_1 \leq X \leq x_2) = P(z_1 \leq Z \leq z_2)\) where \(z_1 = \frac{x_1 - \mu}{\sigma}\) and \(z_2 = \frac{x_2 - \mu}{\sigma}\).

Why do we need to standardize? The purpose for standardization is two-fold:

1. Standardizing allows us to have a more meaningful understanding of the data.
2. When the mean, \(\mu\), or/and variance, \(\sigma^2\) is unknown, standardizing allows us to use the inverse standard normal function in GDC to calculate the unknown normal distribution parameters.

Note: Some advantages of using \(Z \sim N(0,1)\):

1. Symmetry about \(0\).
2. We can express \(Z\) in terms of \(\mu\) and \(\sigma\)
3. If \(P(Z > z_1) = P(Z < z_2)\), we know that \(z_1 = -z_2\). (try sketching the curve to see!)
4. If \(P(Z > z_1) > 0.5\) we can conclude that \(z_1 < 0\).