Conditional expected value

Projection interpretation of conditional expected value

Let 𝑋,𝑌 :𝜉 →ℝ be real random variables. For any function ℎ :ℝ →ℝ, prob

𝔼⁡[(𝑌−𝔼⁡[𝑌∣𝑋])ℎ(𝑋)]=0

implying Corr⁡[𝑌 −𝔼⁡[𝑌 ∣𝑋],ℎ(𝑋)] =0, i.e. 𝑌 −𝔼⁡[𝑌 ∣𝑋] is completely uncorrelated with any function of 𝑋. This motivates the following interpretation of conditional expectation:

Projection interpretation of conditional expected value

𝔼⁡[𝑌 ∣𝑋] contains the entire component of 𝑌 which is a function of 𝑋 (i.e. determined by 𝑋) and the residual component 𝑌 −𝔼⁡[𝑌 ∣𝑋] is completely uncorrelated with any function of 𝑋.

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