Standardization and Cumulative Distribution Function of a Gaussian Random Variable

Let $X$ be a Gaussian random variable on a probability space $(\Omega,\mathcal{F},P)$, with mean $\mu=\mathbb{E}[X]$ and variance $\sigma^{2}=\operatorname{Var}(X)$, both defined and finite by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector. Then:

1. (Degenerate case) If $\sigma^{2}=0$, then $P(X=\mu)=1$.

2. (Standardization) If $\sigma^{2}>0$, then, with $\sigma$ the positive square root of $\sigma^{2}$, the random variable $(X-\mu)/\sigma$ is standard normal, and the cumulative distribution function of $X$ satisfies

where $\Phi$ is the cumulative distribution function of the standard normal distribution as in Standard Normal Distribution.

In particular, the cumulative distribution function of a Gaussian random variable is determined by its mean and variance.