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:

    “If God lived on earth, people would break his windows.”
    —Jewish proverb, quoted in Claud Cockburn, Cockburn Sums Up, epigraph (1981)

    “In the great department store of life, baseball is the toy department.”
    —Los Angeles Sportscaster. quoted in Independent Magazine (London, Sept. 28, 1991)

    “... 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)

    “Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear. We are tending toward the condition of science and aspiring to do it. The artist works out his own formulas; the interest of science lies in the art of making science.”
    —Paul Valéry (1871–1945)