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Given two jointly distributed random variables X and
Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically
calculated by summing or integrating the joint probability
distribution over Y.
For discrete random variables, the marginal probability mass function can be written as
P(X=x). This is
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| P(X = x) = |
∑ |
P(X = x,Y = y) = |
∑ |
P(X = x | Y = y)P(Y = y) |
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y |
|
y |
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where P(X=x,Y=y) is the joint distribution of
X and Y, while P(X=x|Y=y) is the conditional distribution of X given
Y.
Similarly for continuous random variables,
the marginal probability density function
can be written as pX(x). This is
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where pX,Y(x,y) gives the joint distribution of X and Y, while
pX|Y(x|y) gives the conditional distribution for X given
Y.
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