Convolution of probability distributions

The convolution 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.

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 ${\displaystyle Z=X+Y}$ of two independent discrete variables is

The counterpart for independent continuous variables with density functions ${\displaystyle f(x),g(y)}$ is

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