Suppose continuous r.v. XX follows some normal distribution, we denote X∼N(μ,σ2)X\sim \mathcal{N}(\mu,\sigma^2), its pdf is

f(x∣μ,σ2)=12πσ2e−(x−μ)22σ2f(x|\mu,\sigma^2)=\frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

In short, here’s some properties:

  • μX=μ\mu_X=\mu
  • σX2=σ2\sigma_X^2=\sigma^2

Standard Normal Distribution

If X∼N(μ,σ2)X\sim \mathcal{N}(\mu, \sigma^2) and another cont r.v. Z=X−μσZ=\frac{X-\mu}{\sigma}, then Z∼N(0,1)Z\sim \mathcal{N}(0,1).

The pdf of N(0,1)\mathcal{N}(0,1) is denoted as ϕ(⋅)\phi(\cdot), and its cdf is denoted as Φ(⋅)\Phi(\cdot)

In addition, the upper α\alpha-th quantile of N(0,1)\mathcal{N}(0,1), which is the solution to 1−Φ(z)=α1-\Phi(z)=\alpha, is denoted as zαz_\alpha. The α\alpha-th quantile of N(0,1)\mathcal{N}(0,1), i.e. the solution to Φ(z)=α\Phi(z)=\alpha is denoted as Φ−1(α)\Phi^{-1}(\alpha).

The definition directly gives

zα=−Φ−1(α)z_\alpha=-\Phi^{-1}(\alpha)