Probability Density Function - Sums of Independent Random Variables

Sums of Independent Random Variables

See also: Convolution and List of convolutions of probability distributions

The probability density function of the sum of two independent random variables U and V, each of which has a probability density function, is the convolution of their separate density functions:

f_{U+V}(x) = \int_{-\infty}^\infty f_U(y) f_V(x - y)\,dy = \left( f_{U} * f_{V} \right) (x)

It is possible to generalize the previous relation to a sum of N independent random variables, with densities U1, …, UN:

f_{U_{1} + \dotsb + U_{N}}(x) = \left( f_{U_{1}} * \dotsb * f_{U_{N}} \right) (x)

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

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