= Gaussian random variable
{c}
{title2=$X\sim N(\mu,\sigma^2)$}
= Normal random variable
{synonym}
A <Gaussian random variable> is a <random variable> whose <probability distribution> is a <normal distribution>. For $\sigma>0$, write $X=\mu+\sigma Z$ with a <standard normal random variable> $Z$; conversely, <standardization of a normal random variable> recovers $Z=(X-\mu)/\sigma$. For example, $X\sim N(100,15^2)$ has <mean> $100$, <standard deviation> $15$, and <variance> $225$. The outcome $x=115$ corresponds to <standard score> $z=1$, one <standard deviation> above the <mean>.
Some definitions also allow a <Gaussian random variable> with $\sigma=0$. This means $X=\mu$ <almost surely>, with <variance> zero and <probability distribution> equal to the <Dirac measure> $\delta_\mu$. It has no <probability density function> with respect to <Lebesgue measure>: $\mathbb P(X=\mu)=1$. The positive-scale density formula and division by $\sigma$ apply only when $\sigma>0$. MIT's constructive definition explicitly includes the constant case. https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/ceb423b6a703abd60429367b733f45bf_MIT6_436JF18_lec14.pdf[MIT 6.436, Lecture 14, §2: degenerate normal distributions]{ref}.
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