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Concept
Distribution of the product of two random variables
Summary
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product is a product distribution.
The product distribution is the PDF of the product of sample values. This is not the same as the product of their PDF's yet the concepts are often ambiguously termed as "product of Gaussians".
Algebra of random variables
The product is one type of algebra for random variables: Related to the product distribution are the ratio distribution, sum distribution (see List of convolutions of probability distributions) and difference distribution. More generally, one may talk of combinations of sums, differences, products and ratios.
Many of these distributions are described in Melvin D. Springer's book from 1979 The Algebra of Random Variables.
If and are two independent, continuous random variables, described by probability density functions and then the probability density function of is
We first write the cumulative distribution function of starting with its definition
We find the desired probability density function by taking the derivative of both sides with respect to . Since on the right hand side, appears only in the integration limits, the derivative is easily performed using the fundamental theorem of calculus and the chain rule. (Note the negative sign that is needed when the variable occurs in the lower limit of the integration.)
where the absolute value is used to conveniently combine the two terms.
A faster more compact proof begins with the same step of writing the cumulative distribution of starting with its definition:
where is the Heaviside step function and serves to limit the region of integration to values of and satisfying .
We find the desired probability density function by taking the derivative of both sides with respect to .
where we utilize the translation and scaling properties of the Dirac delta function .
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