Editing Probability density function (section)

==Sums of independent random variables==
{{See also|List of convolutions of probability distributions}}

The probability density function of the sum of two [[statistical independence|independent]] random variables {{math|''U''}} and {{math|''V''}}, each of which has a probability density function, is the [[convolution]] of their separate density functions:
<math display="block">
f_{U+V}(x) = \int_{-\infty}^\infty f_U(y) f_V(x - y)\,dy
= \left( f_{U} * f_{V} \right) (x)
</math>

It is possible to generalize the previous relation to a sum of N independent random variables, with densities {{math|''U''<sub>1</sub>, ..., ''U<sub>N</sub>''}}:
<math display="block">f_{U_1 + \cdots + U}(x)
= \left( f_{U_1} * \cdots * f_{U_N} \right) (x)</math>

This can be derived from a two-way change of variables involving {{math|1=''Y'' = ''U'' + ''V''}} and {{math|1=''Z'' = ''V''}}, similarly to the example below for the quotient of independent random variables.