Standard normal distribution

The standard normal distribution is the Normal distribution with mean $𝜇 = 0$ and standard deviation $𝜎 = 𝜎 2 = 1$ .

𝑍 \sim N (0, 1)

It has the simplified probability density function

𝑓 𝑍 (𝑥) = 𝜙 (𝑥) = 1 \sqrt 2 𝜋 𝑒 - 𝑥 2 / 2

Due to its symmetry about $𝑧 = 0$ ,

ℙ (𝑍 < - 𝑧) = ℙ (𝑍 > 𝑧)

Standardisation

Any normally distributed variable $𝑋 \sim N (𝜇, 𝜎 2)$ can be standardised using the Location-scale transformation

𝑍 = 𝑋 - 𝜇 𝜎 \sim N (0, 1)

where the standardised version of a value is sometimes called the Z-score. This allows the calculation of probabilities for any normally-distributed random variable using the Cumulative distribution function for the standard normal distribution $𝐹 𝑍 (𝑧)$ .

ℙ (𝑎 \leq 𝑋 \leq 𝑏) = 𝐹 𝑍 (𝑏 - 𝜇 𝜎) - 𝐹 𝑍 (𝑎 - 𝜇 𝜎)

Properties

Let $𝑍 \sim N (0, 1)$ .

Moment-generating function: $e 𝑡 2 / 2$
Moments: $𝔼 [𝑍 2 𝑛] = (2 𝑛)! 2 𝑛 𝑛!$
Excess kurtosis: 0

#state/tidy | #SemBr