Conditional probability

Conditional variance

Given a random variable

Let 𝑋,𝑌 :𝜉 →ℝ be real random variables. The conditional variance of 𝑌 given 𝑋 is the random variable #m/def/prob

Var⁡[𝑌∣𝑋]=𝔼⁡[𝑌2∣𝑋]−(𝔼⁡[𝑌∣𝑋])2

where we invoke the Conditional expected value.

Properties

  1. Eve's law: Var⁡[𝑌] =𝔼⁡[Var⁡[𝑌 ∣𝑋]] +Var⁡[𝔼⁡[𝑌 ∣𝑋]]

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