Conditional probability

Conditional expected value

Given an event

Let 𝑋 :πœ‰ →ℝ be a real random variable and 𝐴 ∈F be an event with nonzero probability. Then the conditional expected value of 𝑋 given 𝐴 is defined to be prob

𝔼⁑[π‘‹βˆ£π΄]=βˆ«βˆžβˆ’βˆžπ‘¦π‘“π‘Œβˆ£π΄(𝑦)𝑑π‘₯

where π‘“π‘Œβˆ£π΄(𝑦) is the conditional distribution of π‘Œ given 𝐴.

Properties

  • If {𝐴𝑖}𝑛𝑖=1 partition 𝑋 then 𝔼⁑[π‘Œ] =βˆ‘π‘›π‘–=1𝔼⁑[π‘Œ βˆ£π΄π‘–]β„™(𝐴𝑖)

Given a random variable

Let 𝑋,π‘Œ :πœ‰ →ℝ be a real random variables and 𝑔(π‘₯) =𝔼⁑[π‘Œ βˆ£π‘‹ =π‘₯]. Then the conditional expected value of π‘Œ given 𝑋 is the random function

𝔼⁑[π‘Œβˆ£π‘‹]=𝑔(𝑋)

Properties

Let 𝑋,π‘Œ,𝑍,π‘Œ1,π‘Œ2,β‹― :πœ‰ →ℝ, be real random variables.

  1. If 𝑋 and π‘Œ are independent, then 𝔼⁑[π‘Œ βˆ£π‘‹] =𝔼⁑[π‘Œ]
  2. For any function β„Ž :ℝ →ℝ we have 𝔼⁑[β„Ž(𝑋)π‘Œ βˆ£π‘‹] =β„Ž(𝑋)𝔼⁑[π‘Œ βˆ£π‘‹]
  3. Linearity: 𝔼⁑[πœ‡π‘Œ1 +πœ†π‘Œ2 βˆ£π‘‹] =πœ‡[π‘Œ1 βˆ£π‘‹] +πœ†π”Όβ‘[π‘Œ2 βˆ£π‘‹]
  4. Adam’s law: 𝔼⁑[π‘Œ] =𝔼⁑[𝔼⁑[π‘Œ βˆ£π‘‹]]
  5. Adam’s law with extra conditioning: 𝔼⁑[𝔼⁑[π‘Œ βˆ£π‘‹,𝑍] βˆ£π‘] =𝔼⁑[π‘Œ βˆ£π‘]
  6. Projection interpretation of conditional expected value

See also

stub | en | SemBr

Graph View

Backlinks