Expressing the probability of a normally distributed RV with μ,σ.

Question

Given a random variable $X \stackrel{}{\sim}N(μ,σ^2)$, is it possible to express the probability of $X$ being within $y$ standard deviations of the mean solely with $μ and σ?

EX: What is the probability of $X$ being within 2 deviations of it's mean?

Answer 1

Denote by $k$ the number of standard deviations that you are interested in, where $X \sim \mathcal{N}(\mu, \sigma^2)$, thus the probability of being within $k$ standard deviations from the mean $\mu$ is \begin{align} \mathbb{P}(|X - \mu|\le k\sigma ) &= \mathbb{P}(\mu - k\sigma\le X \le \mu + k\sigma )\\ &= \Phi\left( \frac{k\sigma}{\sigma} \right) - \Phi\left( - \frac{ k\sigma}{\sigma} \right) \\ & = 2\Phi\left( k \right) -1. \end{align}

Answer 2

For $a>0$:

$P(|x-\mu|<a\ \sigma) = {\rm Erf}[a/\sqrt{2}]$