Conditional Distributions

The concept of conditional distribution of a random variable combines the concept of distribution of a random variable and the concept of conditional probability.

If we are considering more than one variable, restricting all but one1 of the variables to certain values will give a distribution of the remaining variables.  This is called a conditional distribution.

For example, if we are considering random variables X and Y and 2 is a possible value of X, then we obtain the conditional distribution of Y given X = 2. This conditional distribution is often denoted by Y|(X = 2).

A conditional distribution is a probability distribution, so we can talk about its mean, variance, etc. as we could for any distribution. For example, the conditional mean of the distribution Y|(X = x)  is denoted by E(Y|(X = x).

1. More generally, if we restrict just some of the variables to specific values or ranges, we obtain a joint conditional distribution of the remaining variables. For example, if we consider random variables X, Y, Z and U, then restricting Z and  U to specific values z and u (respectively) gives a conditional joint distribution of X and Y given Z = z and U = u.