Product distribution

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

$Z = XY$

is a product distribution.

1 Algebra of random variables
2 Derivation
3 Special cases
4 See also
5 Notes
6 References

Algebra of random variables

Main article: 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 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.^[1]

Derivation

A way of deriving the product distribution of Z from the joint distribution of the two other random variables, X and Y, is by integration of the following form^[2]

$p_Z(z) = \int^{\infty}_{-\infty} \frac{1}{|x|} \, p_{X,Y}\left(x, \frac{z}{x}\right) \, dx.$

This is not always straightforward.

Special cases

The distribution of the product of two random variables which have lognormal distributions is again lognormal. This is itself a special case of a more general set of results where the logarithm of the product can be written as the sum of the logarithms. Thus, in cases where a simple result can be found in the list of convolutions of probability distributions, where the distributions to be convoluted are those of the logarithms of the components of the product, the result might be transformed to provide the distribution of the product. However this approach is only useful where the logarithms of the components of the product are in some standard families of distributions.

The distribution of the product of a random variable having a uniform distribution on (0,1) with a random variable having a gamma distribution with shape parameter equal to 2, is an exponential distribution.^[3] A more general case of this concerns the distribution of the product of a random variable having a beta distribution with a random variable having a gamma distribution: for some cases where the parameters of the two component distributions are related in a certain way, the result is again a gamma distribution but with a changed shape parameter.^[3]

The K-distribution is an example of a non-standard distribution that can be defined as a product distribution (where both components have a gamma distribution).

Notes

^ Melvin Dale Springer (1979). The Algebra of Random Variables. Wiley. ISBN 0-471-01406-0.
^ Rohatgi, V.K., 1976. An Introduction to Probability Theory Mathematical Statistics. Wiley, New York.
^ ^a ^b Johnson, Noeman L.; Kotz, Samuel; Balakrishnan, N. (1995). Continuous Univariate Distributions Volume 2, Second edition. Wiley. p. 306. ISBN 0-471-58494-0.

References

Springer, MD and Thompson, WE (1970). "The distribution of products of beta, gamma and Gaussian random variables". SIAM Journal on Applied Mathematics 18 (4): 721–737. doi:10.1137/0118065. JSTOR 2099424.
Springer, MD and Thompson, WE (1966). "The distribution of products of independent random variables". SIAM Journal on Applied Mathematics 14 (3): 511–526. doi:10.1137/0114046. JSTOR 2946226.