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)

    “The class of citizens who provide at once their own food and their own raiment, may be viewed as the most truly independent and happy.”
    —James Madison (1751–1836)

    “poor Felix Randal;
    How far from then forethought of, all thy more boisterous years,
    When thou at the random grim forge, powerful amidst peers,
    Didst fettle for the great gray drayhorse his bright and battering
    sandal!”
    —Gerard Manley Hopkins (1844–1889)

    “The variables of quantification, ‘something,’ ‘nothing,’ ‘everything,’ range over our whole ontology, whatever it may be; and we are convicted of a particular ontological presupposition if, and only if, the alleged presuppositum has to be reckoned among the entities over which our variables range in order to render one of our affirmations true.”
    —Willard Van Orman Quine (b. 1908)