Conditional Probability and Conditional Expectation
This domain tests conditional probability as a modelling tool: Bayes' theorem, the law of total probability, conditional expectation as a random variable, and the tower and conditional-variance identities. CS1 frequently embeds these inside multi-stage claims models.
Core rules
- P(A|B)=P(A∩B)/P(B), provided P(B)>0; P(A)=ΣP(A|B_i)P(B_i) for a partition.
- E[X|Y] is a function of Y, hence a random variable. Its expectation is E[X] (tower property).
- Var(X)=E[Var(X|Y)]+Var(E[X|Y]).
- Independence means P(A|B)=P(A); conditional distributions can be used to build joint models.
Worked example
Let NPoisson(λ) be the number of claims and, conditional on N=n, let XBinomial(n,p) be the number paid. Then E[X|N]=Np, so E[X]=E[Np]=λp. For the variance, Var(X|N)=Np(1−p) and E[X|N]=Np. Thus Var(X)=E[Np(1−p)]+Var(Np)=λp(1−p)+λp²=λp. The answer is not λp(1−p): the random number of claims adds variability.
Traps
- Treating E[X|Y] as a fixed number after conditioning; it remains random until Y is observed.
- Conditioning on a null event for continuous variables; use densities or limiting arguments instead.
- Reversing Bayes: P(A|B) is not P(B|A).
- Omitting the between-group variance term in the conditional variance formula.