n. (Statistics) 1. the probability, P(A | B), of one event, A, occurring, given that another, B, is already known to have occurred; this is defined by
P(A | B) = P(A & B)/P(B).
Where x_i and y_j are values of the discrete random variables X and Y, and p_ij is the joint probability of X = x_i and Y = y_j, the conditional probability is given by
P(x_i | y_j) = p_ij/(Σ_i p_ij).
See also Bayes's theorem, Radon-Nikodym theorem. 2. (of a set E in X × Y, given x) formally, the function
μ_(x)(E) = e_(x)(χ_E)
where e_(x)(f) is the conditional expectation, given x, of the random variable; this is defined in terms of the Radon-Nikodym derivative with respect to α, where
α(A) = μ(A × Y)
and χ_E is the characteristic function of E. Here μ_(x) behaves much like a measure, in that, given a countable disjoint family of measurable sets {E_n},
μ_(x)(∪_n E_n) = Σ_n μ_(x)(E_n)
for almost all x with respect to α.