Given two random variables X and Y, the joint distribution of X and Y is the distribution of X and Y together.

The discrete case

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

$P(X=x\backslash \; \backslash mathrm\{and\}\backslash \; Y=y)\; =\; P(Y=y|X=x)P(X=x)=\; P(X=x|Y=y)P(Y=y).\backslash ;$

Since these are probabilities, we have

$\backslash sum\_x\; \backslash sum\_y\; P(X=x\backslash \; \backslash mathrm\{and\}\backslash \; Y=y)\; =\; 1.\backslash ;$

The continuous case

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Similarly for continuous random variables, the joint probability density function can be written as f_X,Y(x, y) and this is

$f\_\{X,Y\}(x,y)=f\_\{Y|X\}(y|x)f\_X(x)\; =\; f\_\{X|Y\}(x|y)f\_Y(y)\; \backslash ;$

where f_Y|X(y|x) and f_X|Y(x|y) give the conditional distributions of Y given X = x and of X given Y = y respectively, and f_X(x) and f_Y(y) give the marginal distributions for X and Y respectively.

Since this is a probability density, we have

$\backslash int\_x\; \backslash int\_y\; f\_\{X,Y\}(x,y)\; \backslash ;\; dy\; \backslash ;\; dx=\; 1.$

Joint distribution of independent variables

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If for discrete random variables $\backslash \; P(X\; =\; x\; \backslash \; \backslash mbox\{and\}\; \backslash \; Y\; =\; y\; )\; =\; P(\; X\; =\; x)\; \backslash cdot\; P(\; Y\; =\; y)$ for all x and y, or for continuous random variables $\backslash \; p\_\{X,Y\}(x,y)\; =\; p\_X(x)\; \backslash cdot\; p\_Y(y)$ for all x and y, then X and Y are said to be independent.

Multidimensional distributions

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The joint distribution of two random variables can be extended to many random variables X₁, ..., X_n by adding them sequentially with the identity

$f\_\{X\_1,\; \backslash ldots,\; X\_n\}(x\_1,\; \backslash ldots,\; x\_n)\; =\; f\_\{X\_n\; |\; X\_1,\; \backslash ldots,\; X\_\{n-1\}\}(\; x\_n\; |\; x\_1,\; \backslash ldots,\; x\_\{n-1\})\; f\_\{X\_1,\; \backslash ldots,\; X\_\{n-1\}\}(\; x\_1,\; \backslash ldots,\; x\_\{n-1\}\; )\; .$

See also

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• Copula (statistics)

External links

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Probability theory