Expectation

The expectation $𝜇 𝑋 = 𝔼 [𝑋] = ⟨ 𝑋 ⟩$ of a Real random variable $𝑋$ may be thought of as the value which the variable is most likely to be close to. It has different but similar definitions for a Discrete variable and a Continuous variable.

Discrete variable

For a Discrete random variable $𝑋$ the expected value $𝔼 [𝑋]$ is defined as follows

𝔼 [𝑋] = \sum 𝑥 \in s u p p [𝑋] 𝑥 𝑝 𝑋 (𝑥)

Expectation value may also be found by summing the Survival function

𝔼 [𝑋] = \infty \sum 𝑛 = 0 ℙ [𝑋 > 𝑛]

Proof

#missing/proof

Continuous variable

For a continuous random variable Continuous random variable $𝑋$ the expected value $𝔼 [𝑋]$ is defined as follows

𝔼 [𝑋] = \int \infty - \infty 𝑥 𝑓 𝑋 (𝑥) 𝑑 𝑥

Properties

The expected value has the following useful properties, where $𝑋$ and $𝑌$ are random variables (possibly dependent) and $𝑎, 𝑏$ are constants.

$𝔼 [𝑎] = 𝑎$
$𝔼 [𝑎 𝑋 + 𝑏 𝑌] = 𝑎 𝔼 [𝑋] + 𝑏 𝔼 [𝑦]$
$𝔼 [𝑋 𝑌] = 𝔼 [𝑋] 𝔼 [𝑌]$ for independently distributed $𝑋, 𝑌$

Proof of 3

Since

𝔼 [𝑋 𝑌] = \int \infty - \infty 𝑥 𝑦 𝑑 ℙ (𝑋 𝑌 = 𝑥 𝑦) = \int \infty - \infty 𝑥 𝑑 ℙ (𝑋 = 𝑥) \int \infty - \infty 𝑦 𝑑 ℙ (𝑌 = 𝑦) = 𝔼 [𝑋] 𝔼 [𝑌]

as required by ^P3.