Gaussian Random Variable

Last updated Nov 1, 2022

Suppose $(\Omega, \mathcal{M}, \mathbb{P})$ is a Probability Space. Then $X$ is a Gaussian Random Variable on $\Omega$ if it has Distribution determined by Probability Density Function

$$f_{X}(x) = \frac{1}{\sqrt{2 \pi} \sigma} e^{- \frac{1}{2} (\frac{x- \mu}{\sigma})^{2}}$$

where $\sigma$ and $\mu$ are parameters for the distribution. We write $X \sim \mathcal{N}(\mu, \sigma^{2})$.

The Probability Density Function is taken with respect to the Lebesgue Measure on $\mathcal{B}(\mathbb{R})$.

1. $\mu = \mathbb{E}(X)$ Proof: TODO
2. $\sigma^{2}= \text{Var}(X)$ Proof: TODO
3. Gaussian Distributions are Symmetric about their Mean