Normal

The Normal (Gaussian) Distribution

The normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is the most important probability distribution in statistics because it accurately describes many natural phenomena and serves as the foundation for inferential statistics.

Definition & Parameters

A random variable $X$ is normally distributed with mean $\mu$ and variance ${\sigma }^2$, denoted as:

$$X\sim \mathcal{N}(\mu ,{\sigma }^2)$$

• Mean ($\mu$): Determines the center of the distribution (where the peak is).
• Standard Deviation ($\sigma$): Determines the spread or width of the bell curve.

Probability Density Function (PDF)

The PDF of a normal distribution is given by:

$$f(x)=\frac{1}{\sigma \sqrt{2\pi }}\exp \left(-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2\right)$$

Theorem (Sum of Independent Normal RVs)

If $X\sim \mathcal{N}({\mu }_X,{\sigma }_X^2)$ and $Y\sim \mathcal{N}({\mu }_Y,{\sigma }_Y^2)$ are independent random variables, then their sum $Z=X+Y$ is also normally distributed:

$$X+Y\sim \mathcal{N}({\mu }_X+{\mu }_Y,{\sigma }_X^2+{\sigma }_Y^2)$$

Variance Summation

Notice that we add the variances (${\sigma }^2$), not the standard deviations ($\sigma$).

For any linear combination $aX+bY$ (where $a$ and $b$ are constants):

$$aX+bY\sim \mathcal{N}(a{\mu }_X+b{\mu }_Y,a^2{\sigma }_X^2+b^2{\sigma }_Y^2)$$

Theorem (Standardization)

Any normal random variable $X$ can be transformed into a Standard Normal Distribution ($\mathcal{N}(0,1)$) using the $Z$-score formula:

$$Z=\frac{X-\mu }{\sigma }$$

Central Limit Theorem (CLT)

The Central Limit Theorem states that for a sufficiently large sample size ($n\ge 30$), the distribution of the sample mean $\overline{X}$ will be approximately normal, even if the underlying population is not:

$$\overline{X}\approx \mathcal{N}\left(\mu ,\frac{{\sigma }^2}{n}\right)$$