2.2c.3 Gaussian Distribution

Gaussian Distribution

A continuous random variable $X\in \mathbb{R}$ follows the Gaussian distribution $X\sim \mathcal{N}(\mu ,{\sigma }^2)$ with parameters $\mu \in \mathbb{R}$ and variance ${\sigma }^2>0$, defined by the probability density function
$f_X(x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-(x-\mu {)}^2/2{\sigma }^2}.$

The Gaussian distribution is used to model real-valued data that tend to cluster around a central value (mean) $\mu$ with symmetric dispersion $\sigma$.

Moreover, if $X\sim \mathcal{N}(0,1)$, then $X$ is a standard normal random variable. Its probability density function simplifies to $f_X(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$, which leads to its cumulative distribution function,
$\Phi (x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^x{e}^{-x^2/2}dx.$

A key characteristic of the normal distribution is the empirical rule (68-95-99 rule), which states that around $68\%$, $95\%$, and $99\%$ of the data falls within $1\sigma$, $2\sigma$, and $3\sigma$ of the mean.

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

Any Gaussian random variable can be standardized into the Normal distribution using the transformation

Properties

(Mean) $\mathbb{E}[X\sim \mathcal{N}(\mu ,{\sigma }^2)]=\mu$.

(Variance) $Var[X\sim \mathcal{N}(\mu ,{\sigma }^2)]={\sigma }^2$.

(Sum) If $X\sim \mathcal{N}(\mu ,{\sigma }_X^2)$ and $Y\sim \mathcal{N}(\eta ,{\sigma }_Y^2)$, then $X+Y\sim \mathcal{N}(\mu +\eta ,{\sigma }_X^2+{\sigma }_Y^2)$.

(Rotational Symmetry)

(Positive Gaussian) $\mathbb{E}[X\mid X>0]=\sigma \frac{1}{\sqrt{2\pi }}$ and $\mathbb{E}[X\mid X+Y>0]$.

Since we know that $f_{X\mid A}(x)=\frac{1}{\mathbb{P}(X\in A)}$, we proceed as follows:

\mathbb{E}[X \mid X > 0] &= \int^{\infty}_{0} xf_{X \mid X >0}(x)dx \\ &= \frac{\int^{\infty}_{ 0 }xf_{X}(x)dx}{\mathbb{P}(X>0)} \\ &= \frac{\int^{\infty}_{ 0 }xf_{X}(x)dx}{0.5} \\ &= \frac{2}{\sigma\sqrt{ 2\pi }}\int^{\infty}_{ 0 }xe^{-x^2/2\sigma^2}dx \\ &= -\sigma^2\sqrt{ \frac{2}{\pi\sigma^2} } e^{-x^2/2\sigma^2} \bigg|^{\infty}_{x=0} \\ &= \sigma \sqrt{ \frac{2}{\pi} } \end{align}$$ # Examples