Normal Distribution (gaussian)

3.5 Normal Distribution and Probability

Normal (Gaussian) Distribution

• A normal random variable $X\sim \mathcal{N}(\mu ,{\sigma }^2)$ has probability-density function (PDF) $$f(x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp [-\frac{(x-\mu {)}^2}{2{\sigma }^2}],-\infty <x<\infty .$$
• Standardisation: the $z$-score
  $z=\frac{x-\mu }{\sigma }⟹Z\sim \mathcal{N}(0,1).$

Cumulative-Distribution Function (CDF) of a Gaussian

• The CDF gives the probability that $X$ does not exceed a threshold: $$F_X(x)=P(X\le x)=\frac{1}{2}[1+\mathrm{erf}(\frac{x-\mu }{\sigma \sqrt{2}}\left)\right],$$ where $\mathrm{erf}$ is the error function.
• In practice we consult the standard-normal table (or software) for $P(Z\le z)$ and convert $x\to z$ using the formula above.