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Given two jointly distributed random variables X and
Y, the conditional distribution of Y given X (written "Y|X") is the
probability distribution of Y when
X is known to be a particular value.
For discrete random variables, the conditional probability mass function can be written as
P(Y=y|X=x). From Bayes'
theorem, this is
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Similarly for continuous random variables,
the conditional probability density
function can be written as pY|X(y|x) and this is
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where pX,Y(x,y) gives the joint distribution of
X and Y, while pX(x) gives the marginal distribution for X.
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.
If for discrete random variables P(Y=y|X=x)=P(Y=y) for
all x and y, or for continuous random variables
pY|X(y|x)=pY(y) for all x and y,
then Y is said to be independent of
X (and this implies that X is also independent of 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 density) is 1. Seen as a function of x for given
y, it is a likelihood, so that the sum over all x need not be
1.
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