Editing Joint probability distribution (section)

===Discrete case===
The joint [[probability mass function]] of two [[discrete random variable]]s <math>X, Y</math> is:

{{Equation box 1
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|equation = {{NumBlk||<math>p_{X,Y}(x,y) = \mathrm{P}(X=x\ \mathrm{and}\ Y=y)</math>|{{EquationRef|Eq.3}}}}
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or written in terms of conditional distributions
:<math>p_{X,Y}(x,y) = \mathrm{P}(Y=y \mid X=x) \cdot \mathrm{P}(X=x) = \mathrm{P}(X=x \mid Y=y) \cdot \mathrm{P}(Y=y)</math>
where <math> \mathrm{P}(Y=y \mid X=x) </math> is the [[conditional probability|probability]] of <math> Y = y </math> given that <math> X = x </math>.

The generalization of the preceding two-variable case is the joint probability distribution of <math>n\,</math> discrete random variables <math>X_1, X_2, \dots,X_n</math> which is:

{{Equation box 1
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|equation = {{NumBlk||<math>p_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = \mathrm{P}(X_1=x_1\text{ and }\dots\text{ and }X_n=x_n)</math>|{{EquationRef|Eq.4}}}}
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or equivalently

:<math>
\begin{align}
p_{X_1,\ldots,X_n}(x_1,\ldots,x_n) & =  \mathrm{P}(X_1=x_1) \cdot \mathrm{P}(X_2=x_2\mid X_1=x_1) \\ & \cdot \mathrm{P}(X_3=x_3\mid X_1=x_1,X_2=x_2)  \\ &  \dots \\  & \cdot P(X_n=x_n\mid X_1=x_1,X_2=x_2,\dots,X_{n-1}=x_{n-1}).
\end{align}
</math>.

This identity is known as the [[Chain rule (probability)|chain rule of probability]].

Since these are probabilities, in the two-variable case

:<math>\sum_i \sum_j \mathrm{P}(X=x_i\ \mathrm{and}\ Y=y_j) = 1,\,</math>
which generalizes for <math>n\,</math> discrete random variables <math>X_1, X_2, \dots , X_n</math> to

:<math>\sum_{i} \sum_{j} \dots \sum_{k} \mathrm{P}(X_1=x_{1i},X_2=x_{2j}, \dots, X_n=x_{nk}) = 1.\;</math>