Standard Normal

In statistics, the term "standard normal" usually refers to a specific type of normal distribution, also known as the standard normal distribution or the Gaussian distribution with mean 0 and variance 1. The symbol $Z$ denotes it.

The PDF of the standard normal distribution is given by:

$$f(z) = \frac{1}{\sqrt{2\pi}}\cdot e^{-\frac{z^2}{2}}$$ $$z = {{x - \mu} \over \sigma}$$

where $z$ is a real number, $\pi$ is the mathematical constant π, and $e$ is the mathematical constant e.

The cumulative distribution function (CDF) of the standard normal distribution, denoted by $\Phi(z)$, gives the probability that a random variable from this distribution is less than or equal to a specific value $z$. It is defined as:

$$\Phi(z) = \int\_{-\infty}^{z}\phi(u)du$$

Where $\Phi(u)$ is the PDF of the standard normal distribution.

The standard normal distribution is essential in statistics because it allows us to standardize other normal distributions with arbitrary means and standard deviations. By converting a normal distribution with mean $\mu$ and standard deviation $\sigma$ to a standard normal distribution with mean 0 and variance 1, we can compare observations from different normal distributions on a standard scale. This standardization process is known as the z-score transformation.