The Gaussian Distribution, also known as the *Normal Distribution*, describes an unbounded [[Random Variable]] $X$, meaning it can take any value in $(-\infty, \infty)$. This is because the exponential term $e^x$ in the [[Probability Density Function|PDF]] never equals zero. However, the Gaussian has *rapidly decaying tails*, meaning that for values of $x$ far from the mean $\mu$, the probability density becomes negligible. **Probability Density (PDF):** $ f_{\mu, \sigma^2}(x)=\frac{1}{\sigma \sqrt{2 \pi}}*\exp\Big(-\frac{(x- \mu)^2}{2 \sigma^2} \Big) $ **Cumulative Density (CDF):** $ F_{\mu, \sigma^2}(x)=\frac{1}{\sigma \sqrt{2 \pi}}* \int_{- \infty}^x \exp\Big(-\frac{(x- \mu)^2}{2 \sigma^2} \Big) dx $ Note that the integral for the [[Cumulative Density Function|CDF]] cannot be solved, since there is no close-form. We rely on approximations from tables or computers. ## Properties of the Gaussian Distribution **Affine transformation:** Denotes the process of adding constants, or multiplying by factors. The Gaussian is invariant to these kind of transformations, meaning the affine transformation of a Gaussian returns a Gaussian. ^e84e63 $ \begin{align} \text{if : }&X &&\sim \mathcal N(\mu, \sigma^2)\\ \text{then : }& a*X+ b &&\sim \mathcal N(a\mu+b, a^2 \sigma^2) \end{align} $ **Standardization:** By subtracting the mean, and dividing by standard deviation, we can transform any normal distribution into a standard normal $\mathcal N(0,1)$. $ \begin{align} \text{if : }&X &&\sim \mathcal N(\mu, \sigma^2)\\ \text{then : }&Z = \frac{X-\mu}{\sigma} &&\sim \mathcal N(0,1) \end{align} $ **Symmetry:** A Gaussian distribution is symmetric around its mean. When it is centered around zero $(\mu=0)$, multiplying $X$ by $-1$ results in the same distribution. ^4ff648 $ \begin{align} \text{if : } &X &&\sim \mathcal N(0, \sigma^2)\\ \text{then : } &-X&&\sim\mathcal N(0, \sigma^2) \end{align} $ Additionally, the symmetry allows the following relationship for probabilities of absolute values: $ \begin{aligned} \mathbf P(\lvert X \rvert>x) &=\mathbf P(X>x) + \mathbf P(X< -x) \\ &=\mathbf P(X>x) + \mathbf P(-X > x) \\ &=2 \mathbf P(X>x) \end{aligned} $ ## Quantiles The quantile function relates probabilities to specific values of $X$. For a given quantile $q_\alpha$ the probability that $X$ is *greater than or equal* to $q_\alpha$ is $\alpha$. More intuitively we think of the reverse case where $X <q_\alpha$. $ \begin{align} \mathbf P(X \ge q_\alpha)&= \alpha \\ \mathbf P(X\le q_\alpha)&=1- \alpha \end{align} $ For example the 90th quantile corresponds to $\alpha=0.1$. $ \begin{align} \mathbf P(X\ge q_{10})&=0.1 \\ \mathbf P(X\le q_{10})&=0.9 \end{align} $ To compute quantiles, we use the inverse CDF (also called the quantile function): $q_\alpha=F^{-1}(1-\alpha)$ This involves finding the $x$-value corresponding to a given cumulative probability $1 - \alpha$. In practice, tables or numerical methods are used to evaluate $F^{-1}$. $ \begin{aligned} F(q_\alpha)&= 1 - \alpha \\ F^{-1}(1- \alpha) &= q_\alpha \end{aligned} $