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Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value. If the conditional distribution of Y given X is a continuous distribution, then its probability density function is known as the conditional density function.
As seen from the definition, and due to its occurrence, it is necessary that
The relation with the probability distribution of X given Y is:
The relation with the probability distribution of X given Y is given by:
The concept of the conditional distribution of a continuous random variable is not as intuitive as it might seem: Borel's paradox shows that conditional probability density functions need not be invariant under coordinate transformations.
Relation to independence
Random variables X, Y are independent if and only if the conditional distribution of Y given X is equal to the unconditional distribution of Y. For discrete random variables: P(Y = y | X = x) = P(Y = y) for all relevant x and y. For continuous random variables having a joint density: fY(y | X=x) = fY(y) for all relevant x and y.
Seen as a function of y for given x, P(Y = y | X = x) is a probability and so the sum over all y (or integral if it is a conditional probability density) is 1. Seen as a function of x for given y, it is a likelihood function, so that the sum over all x need not be 1.
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