Editing Probability density function (section)

== Densities associated with multiple variables ==<!-- This section is linked from [[Sufficiency (statistics)]] -->

For continuous [[random variable]]s {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}}, it is also possible to define a probability density function associated to the set as a whole, often called '''joint probability density function'''. This density function is defined as a function of the {{mvar|n}} variables, such that, for any domain {{mvar|D}} in the {{mvar|n}}-dimensional space of the values of the variables {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}}, the probability that a realisation of the set variables falls inside the domain {{mvar|D}} is
<math display="block">\Pr \left( X_1,\ldots,X_n \isin D \right)
 = \int_D f_{X_1,\ldots,X_n}(x_1,\ldots,x_n)\,dx_1 \cdots dx_n.</math>

If {{math|1=''F''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) = Pr(''X''<sub>1</sub> ≤ ''x''<sub>1</sub>, ..., ''X''<sub>''n''</sub> ≤ ''x''<sub>''n''</sub>)}} is the [[cumulative distribution function]] of the vector {{math|(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}}, then the joint probability density function can be computed as a partial derivative
<math display="block"> f(x) = \left.\frac{\partial^n F}{\partial x_1 \cdots \partial x_n} \right|_x </math>

===Marginal densities===
For {{math|1=''i'' = 1, 2, ..., ''n''}}, let {{math|''f''<sub>''X''<sub>''i''</sub></sub>(''x''<sub>''i''</sub>)}} be the probability density function associated with variable {{math|''X<sub>i</sub>''}} alone.  This is called the marginal density function, and can be deduced from the probability density associated with the random variables {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} by integrating over all values of the other {{math|''n'' − 1}} variables:
<math display="block">f_{X_i}(x_i) = \int f(x_1,\ldots,x_n)\, dx_1 \cdots dx_{i-1}\,dx_{i+1}\cdots dx_n .</math>

===Independence===

Continuous random variables {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} admitting a joint density are all [[statistical independence|independent]] from each other if
<math display="block">f_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = f_{X_1}(x_1)\cdots f_{X_n}(x_n).</math>

===Corollary===

If the joint probability density function of a vector of {{mvar|n}} random variables can be factored into a product of {{mvar|n}} functions of one variable
<math display="block">f_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = f_1(x_1)\cdots f_n(x_n),</math>
(where each {{math|''f<sub>i</sub>''}} is not necessarily a density) then the {{mvar|n}} variables in the set are all [[statistical independence|independent]] from each other, and the marginal probability density function of each of them is given by
<math display="block">f_{X_i}(x_i) = \frac{f_i(x_i)}{\int f_i(x)\,dx}.</math>

===Example===

This elementary example illustrates the above definition of multidimensional probability density functions in the simple case of a function of a set of two variables. Let us call <math>\vec R</math> a 2-dimensional random vector of coordinates {{math|(''X'', ''Y'')}}: the probability to obtain <math>\vec R</math> in the quarter plane of positive {{math|''x''}} and {{math|''y''}} is
<math display="block">\Pr \left( X > 0, Y > 0 \right)
 = \int_0^\infty \int_0^\infty f_{X,Y}(x,y)\,dx\,dy.</math>