Convolution of probability distributions
The convolution/sum of probability distributions arises in probability theory and statistics as the operation in terms of probability distributions that corresponds to the addition of independent random variables and, by extension, to forming linear combinations of random variables. The operation here is a special case of convolution in the context of probability distributions. Introduction. The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of random  variables is the convolution of their corresponding probability mass functions or probability density functions respectively. Many well known distributions have simple convolutions: see List of convolutions of probability distributions The general formula for the distribution of the sum formula_1 of two independent integer-valued (and hence discrete) random variables is For independent, continuous random variables with pdf formula_3 and CDF formula_4 respectively, we have that the CDF of the sum is: If we start with random variables X and Y, related by Z=X+Y, and without knowledge of these random variables being independent, then: However, if X and Y  are independent, then: and this formula becomes the convolution of probability distributions: Example derivation. There are several ways of deriving formulae for the convolution of probability distributions. Often the manipulation of integrals can be avoided by use of some type of generating function. Such methods can also be useful in deriving properties of the resulting distribution, such as moments, even if an explicit formula for the distribution itself cannot be derived. One of the straightforward techniques is to use characteristic functions, which always exists and are unique to a given distribution. Convolution of Bernoulli distributions. The convolution of two  independent identically distributed Bernoulli random variables is a binomial random variable. That is, in a shorthand notation, To show this let Also, let "Z" denote a generic binomial random variable: Using probability mass functions. As formula_12 are independent, Here, we used the fact that formula_14 for "k">"n" in the last but three equality, and of Pascal's rule in the second last equality. Using characteristic functions. The characteristic function of each formula_15 and of formula_16 is where "t" is within some neighborhood of zero. The expectation of the product is the product of the expectations since each formula_15 is independent . Since formula_20 and formula_16 have the same characteristic function, they must have the same distribution.