Normal distribution

The normal distribution or bell-curve is occurs frequently over a diverse range of applications.

𝑋 \sim N (𝜇, 𝜎 2)

𝑓 𝑋 (𝑥) = 1 \sqrt 2 𝜋 𝜎 2 𝑒 - (𝑥 - 𝜇) 2 / 2 𝜎 2

with no closed-form Cumulative distribution function. One important property of the normal distribution is that it is symmetric about $𝜇$ .

ℙ (𝑍 < 𝜇 - 𝑎) = ℙ (𝑍 > 𝜇 + 𝑎)

Graph

Properties

Let $𝑋 \sim N (𝜇, 𝜎 2)$ .

Expectation: $𝔼 [𝑋] = 𝜇$
Variance: $V a r [𝑋] = 𝜎 2$
Moment-generating function: $𝑀 𝑋 (𝑡) = 𝑀 𝜇 + 𝜎 𝑍 (𝑡) = e x p (𝜇 𝑡 + 12 𝜎 2 𝑡 2)$

Furthermore

If $𝑋 1 \sim N (𝜇 1, 𝜎 21)$ and $𝑋 2 \sim N (𝜇 2, 𝜎 22)$ are independently distributed then $𝑋 1 + 𝑋 2 \sim N (𝜇 1 + 𝜇 2, 𝜎 21 + 𝜎 22)$ .
By Cramér's theorem the converse of the above also holds.

Proof of 1

𝑀 𝑋 1 + 𝑋 2 (𝑡) = 𝑀 𝑋 1 (𝑡) 𝑀 𝑋 2 (𝑡) = e x p (𝜇 1 𝑡 + 12 𝜎 21 𝑡 2) e x p (𝜇 2 𝑡 + 12 𝜎 22 𝑡 2) = e x p ((𝜇 1 + 𝜇 2) 𝑡 + 12 (𝜎 21 + 𝜎 22) 𝑡 2)

as required.

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