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Encyclopedia > 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. Probability theory is the mathematical study of probability. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... This article defines some terms which characterize probability distributions of two or more variables. ...


For discrete random variables, the marginal probability mass function can be written as Pr(X = x). This is In mathematics, a random variable is discrete if its probability distribution is discrete; a discrete probability distribution is one that is fully characterized by a probability mass function. ... This article defines some terms which characterize probability distributions of two or more variables. ...

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

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. Given two random variables X and Y, the joint probability distribution of X and Y is the probability distribution of X and Y together. ... Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X (written Y | X) is the probability distribution of Y when X is known to be a particular value. ...


Similarly for continuous random variables, the marginal probability density function can be written as pX(x). This is By one convention, a random variable X is called continuous if its cumulative distribution function is continuous. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

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

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.


Why the name 'marginal'? One explanation is to imagine the p(x,y) in a 2D table such as a spreadsheet. The marginals are obtained 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.


  Results from FactBites:
 
Marginal distribution - Wikipedia, the free encyclopedia (212 words)
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.
Y = y) is the conditional distribution of X given Y.
The marginals are obtained by summing the columns (or rows) -- the column sum would then be written in the margin of the table, ie.
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