Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by a bell-shaped curve, where data clusters symmetrically around its mean, with a majority falling within one standard deviation, making it a fundamental concept in statistics and probability theory.

A normal random variable is denoted:

\[ X \sim \text{Normal}(\mu, \sigma) \]

A standard normal distribution has a mean (\(\mu\)) of \(0\) and standard deviation (\(\sigma\)) of \(1\).

\[ Z \sim \text{Normal}(0,1) \]

Probability Distribution Function (PDF)

The pdf of the standard normal distribution is given as followed:

\[ f_Z(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \text{ if } z \in \mathbb{R} \]

The pdf of the normal distribution is:

\[ f_X(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2} \left(\frac{x-\mu}{\sigma} \right)^2} \text{ if } x \in \mathbb{R} \]

Expectation

The expected value of the standard normal distribution is clearly \(0\), bet we can derive it here:

\[\begin{split} \begin{align} EZ &= \int_\infty^\infty z f_Z(z) dz \\ &= \frac{1}{\sqrt{2\pi}} \int_\infty^\infty z e^{-\frac{z^2}{2}} dz \\ \end{align} \end{split}\]

Note that \( z e^{-\frac{z^2}{2}} \) is an odd function, so it is symmetrical and the mean is \(0\).