## Conditional Probability Distributions

Given two jointly distributed random variables and the conditional probability distribution of given is the probability distribution of when is known to be a particular value. 0 1 2  0 0.05 0.05 0 0.1 1 0.2 0.05 0.15 0.4 2 0.1 0.06 0.04 0.2 3 0.1 0.08 0.12 0.3 0.45 0.24 0.31

1The conditional probabilities for given are obtained by dividing each entry by the righthandmost entry shown in bold, giving 0 1 2  0   0.1 1   0.4 2   0.2 3   0.3 0.45 0.24 0.31

1Note that each row sums to one.

The conditional probabilities for given are obtained by dividing each entry by the bottom entry in the column shown in bold, giving 0 1 2  0   0.1 1   0.4 2   0.2 3   0.3 0.45 0.24 0.31 1

Note that each column sums to one.

To generalise, for discrete random variables, the conditional probability mass function of given (the occurrence of) the value of with can be written, using the definition of conditional probability, as: We can write down also the probability distribution of given  From these we deduce (1)

Similarly for continuous random variables, the conditional probability density function X given the value y of Y is and the conditional probability density function of given the value of can be written as where gives the joint density of and while for gives the marginal distribution function for Similarly as for (1) we can write If for discrete random variables for all and or for continuous random variables or or equivalently for all and then and are independent.

As a function of given is a probability and so the sum over all (or integral if it is a conditional probability density) is 1. Seen as a function of for given it is a likelihood function, so that the sum over all need not be 1. 