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In the study of probability, given two random variables X and Y, the joint distribution of X and Y is the distribution of X and Y together.
The discrete caseEdit
For discrete random variables, the joint probability mass function can be written as Pr(X = x & Y = y). This is
Since these are probabilities, we have
The continuous caseEdit
Similarly for continuous random variables, the joint probability density function can be written as fX,Y(x, y) and this is
where fY|X(y|x) and fX|Y(x|y) give the conditional distributions of Y given X = x and of X given Y = y respectively, and fX(x) and fY(y) give the marginal distributions for X and Y respectively.
Again, since these are probability distributions, one has
Joint distribution of independent variables Edit
If for discrete random variables for all x and y, or for continuous random variables for all x and y, then X and Y are said to be independent.
The joint distribution of two random variables can be extended to many random variables X1, ..., Xn by adding them sequentially with the identity