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.

Read more about this topic:  Probability Density Function

Famous quotes containing the words sums, independent, random and/or variables:

    “At Timon’s villalet us pass a day,
    Where all cry out,What sums are thrown away!’”
    —Alexander Pope (1688–1744)

    “Most works of art are effectively treated as commodities and most artists, even when they justly claim quite other intentions, are effectively treated as a category of independent craftsmen or skilled workers producing a certain kind of marginal commodity.”
    —Raymond Williams (1921–1988)

    “... the random talk of people who have no chance of immortality and thus can speak their minds out has a setting, often, of lights, streets, houses, human beings, beautiful or grotesque, which will weave itself into the moment for ever.”
    —Virginia Woolf (1882–1941)

    “The variables are surprisingly few.... One can whip or be whipped; one can eat excrement or quaff urine; mouth and private part can be meet in this or that commerce. After which there is the gray of morning and the sour knowledge that things have remained fairly generally the same since man first met goat and woman.”
    —George Steiner (b. 1929)