Theory

Theory 1

Conditional distribution - fixed event

Suppose is a random variable, and suppose . The distribution of conditioned on describes the probabilities of values of given knowledge that .

Discrete case:

Continuous case:

There is also a conditional CDF, of which this conditional PDF is the derivative:

The Law of Total Probability has versions for distributions:

Conditional distribution - variable event

Suppose and are any two random variables. The distribution of conditioned on describes the probabilities of values of in terms of , given knowledge that .

Discrete case:

Continuous case:

Remember: is the probability that “ and .”

Sometimes it is useful to have the formulas rewritten like this:

Extra - Deriving

The density ought to be such that gives the probability of , given knowledge that . Calculate this probability: