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Joint probability distribution
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<p>Given <a href="/facts/Random_variable/TwTBXnLT">random variables</a> X , Y , … {\displaystyle X,Y,\ldots } , that are defined on the same <a href="/facts/Probability_space/dEcv2VrU">probability space</a>, the multivariate or joint probability distribution for X , Y , … {\displaystyle X,Y,\ldots } is a <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> that gives the probability that each of X , Y , … {\displaystyle X,Y,\ldots } falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a <a href="/facts/Bivariate_distribution/klX2ksGY">bivariate distribution</a>, but the concept generalizes to any number of random variables. </p><p>The joint probability distribution can be expressed in terms of a joint <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> and either in terms of a joint <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> (in the case of <a href="/facts/Continuous_variable/TWl5feEw">continuous variables</a>) or joint <a href="/facts/Probability_mass_function/LhurokRt">probability mass function</a> (in the case of <a href="/facts/Discrete_probability_distribution/EpsKKVRu">discrete</a> variables). These in turn can be used to find two other types of distributions: the <a href="/facts/Marginal_density/U9XBWAd1">marginal distribution</a> giving the probabilities for any one of the variables with no reference to any specific ranges of values for the other variables, and the <a href="/facts/Conditional_probability_distribution/0eGm3P9W">conditional probability distribution</a> giving the probabilities for any subset of the variables conditional on particular values of the remaining variables. </p>