probability conditioning on a sigma
algebra∗
CWoo†
2013-03-21 21:30:26
Let (Ω, B, µ) be a probability space and B ∈ B an event. Let D be a sub
sigma algebra of B. The conditional probability of B given D is defined to be the
conditional expectation of the random variable 1B defined on Ω, given D. We
denote this conditional probability by µ(B|D) := E(1B |D). 1B is also known
as the indicator function.
Similarly, we can define a conditional probability given a random variable.
Let X be a random variable defined on Ω. The conditional probability of B
given X is defined to be µ(B|BX ), where BX is the sub sigma algebra of B,
generated by X. The conditional probability of B given X is simply written
µ(B|X).
Remark. Both µ(B|D) and µ(B|X) are random variables, the former is
D-measurable, and the latter is BX -measurable.
∗ hProbabilityConditioningOnASigmaAlgebrai created: h2013-03-21i by: hCWooi version:
h38568i Privacy setting: h1i hDefinitioni h60A99i h60A10i
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