Marginal distribution

In probability theory, 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.

Related Topics:
Probability theory - Random variable - Probability distribution - Joint probability

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For discrete random variables, the marginal probability mass function can be written as Pr(X = x). This is

Related Topics:
Discrete random variable - Marginal probability

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:Pr(X=x) = sum_{y} Pr(X=x,Y=y) = sum_{y} Pr(X=x|Y=y) Pr(Y=y),

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where Pr(X = x,Y = y) is the joint distribution of X and Y, while Pr(X = x|Y = y) is the conditional distribution of X given Y.

Related Topics:
Joint distribution - Conditional distribution

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Similarly for continuous random variables, the marginal probability density function can be written as pX(x). This is

Related Topics:
Continuous random variable - Probability density function

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:p_{X}(x) = int_y p_{X,Y}(x,y) , dy = int_y p_{X|Y}(x|y) , p_Y(y) , dy

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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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Why the name 'marginal'? One explanation is to imagine the p(x,y) in a 2D table such as a spreadsheet. The marginals are got by summing the columns (or rows) -- the column sum would then be written in the margin of the table, ie. the column at the side of the table.

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