Conditional Probability

Summary

18.1 Let (Ω, ℱ, P) be a probability space and let $\mathcal{D}$ be a sub-σ-algebra of ℱ. For any random variable Y from (Ω, ℱ) into R^k, there exists a unique (up to a.e. equality) random variable Z from (Ω, $\mathcal{D}$) into R^k such that ∫ _A ZdP=∫ _A YdP for all A in $\mathcal{D}$. Z is called the conditional expected value of Y given 풟. If Y is of order 2, Z is the orthogonal projection of Y on a closed subspace of L²(Ω, P; R^k) (Theorem 18.1.2).

18.2 In this section we prove a measure theoretic converse of the mean value theorem.

18.3 Here, we generalize Jensen’s inequality to conditional expected values: if φ is convex and Y is P-integrable, φ∘ E(Y|풟)≤E(φ(Y) $\mathcal{D}$) (Proposition 18.3.1).

18.4 We define the conditional expected value of Y given the random variable X. When the law of X is absolutely continuous with respect to Lebesgue measure, we can compute E(Y|X) (as a limit) outside some P _X -negligible set (Proposition 18.4.1).

18.5 This section is devoted to the study of the conditional law of Y given X.

18.6 We compute some conditional laws when P _X is defined by masses and when P_[X,Y] is absolutely continuous with respect to a product μ⨂ν.

18.7 We prove the existence of conditional laws when Y is an R^k valued random variable (Theorem 18.7.1).