Conditional expected value

Given an event

Let $𝑋 : 𝜉 \to ℝ$ be a real random variable and $𝐴 \in F$ be an event with nonzero probability. Then the conditional expected value of $𝑋$ given $𝐴$ is defined to be #m/def/prob

𝔼 [𝑋 ∣ 𝐴] = \int \infty - \infty 𝑦 𝑓 𝑌 ∣ 𝐴 (𝑦) 𝑑 𝑥

where $𝑓 𝑌 ∣ 𝐴 (𝑦)$ is the conditional distribution of $𝑌$ given $𝐴$ .

Properties

If ${𝐴 𝑖} 𝑛 𝑖 = 1$ partition $𝑋$ then $𝔼 [𝑌] = \sum 𝑛 𝑖 = 1 𝔼 [𝑌 ∣ 𝐴 𝑖] ℙ (𝐴 𝑖)$

Given a random variable

Let $𝑋, 𝑌 : 𝜉 \to ℝ$ be a real random variables and $𝑔 (𝑥) = 𝔼 [𝑌 ∣ 𝑋 = 𝑥]$ . Then the conditional expected value of $𝑌$ given $𝑋$ is the random function

𝔼 [𝑌 ∣ 𝑋] = 𝑔 (𝑋)

Properties

Let $𝑋, 𝑌, 𝑍, 𝑌 1, 𝑌 2, \dots : 𝜉 \to ℝ,$ be real random variables.

If $𝑋$ and $𝑌$ are independent, then $𝔼 [𝑌 ∣ 𝑋] = 𝔼 [𝑌]$
For any function $ℎ : ℝ \to ℝ$ we have $𝔼 [ℎ (𝑋) 𝑌 ∣ 𝑋] = ℎ (𝑋) 𝔼 [𝑌 ∣ 𝑋]$
Linearity: $𝔼 [𝜇 𝑌 1 + 𝜆 𝑌 2 ∣ 𝑋] = 𝜇 [𝑌 1 ∣ 𝑋] + 𝜆 𝔼 [𝑌 2 ∣ 𝑋]$
Adam's law: $𝔼 [𝑌] = 𝔼 [𝔼 [𝑌 ∣ 𝑋]]$
Adam's law with extra conditioning: $𝔼 [𝔼 [𝑌 ∣ 𝑋, 𝑍] ∣ 𝑍] = 𝔼 [𝑌 ∣ 𝑍]$
Projection interpretation of conditional expected value

Conditional expected value

Given an event

Properties

Given a random variable

Properties

See also