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# 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 one^{1}
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.