Product distribution

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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.

Algebra of random variables[edit]

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.[1]

Derivation for independent random variables[edit]

If and are two independent, continuous random variables, described by probability density functions and then the probability density function of is[2]


Proof [3][edit]

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.

Alternate Proof[edit]

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 .

A more intuitive description of the procedure is illustrated in the figure below. The joint pdf exists in the x-y plane and an arc of constant z value is shown as the shaded line. To find the marginal probability on this arc, integrate over increments of area on this contour.

Diagram to illustrate the product distribution of two variables.

We have so the increment of probability is . If can be equated with , then integration over x, yields yields the integral above.

A Bayesian Interpretation[edit]

Let be a random sample drawn from probability distribution . Scaling by generates a sample from scaled distribution which can be written as a conditional distribution .

Letting be a random variable with pdf , the distribution of the scaled sample becomes and integrating out we get so is drawn from this distribution . However, substituting the definition of we also have which has the same form as the product distribution above. Thus the Bayesian posterior distribution is the distribution of the product of the two independent random samples and .


For the case of one variable being discrete, let . The conditional density is . Therefore .

Expectation of product of random variables[edit]

When two random variables are statistically independent, the expectation of their product is the product of their expectations. This can be proved from the Law of total expectation:

In the inner expression, Y is a constant. Hence:

This is true even if X and Y are statistically dependent. However, in general is a function of Y. In the special case in which X and Y are statistically independent, it is a constant independent of Y. Hence:

Characteristic function of product of random variables[edit]

Assume X, Y are independent random variables. The characteristic function of X is , and the distribution of Y is known. Then from the Law of total expectation, we have

If the characteristic functions and distributions of both X and Y are known, then alternatively, also holds.

Mellin Transform[edit]

The Mellin transform of a distribution with support only on and having a random sample is

The inverse transform is

if are two independent random samples from different distributions, then the Mellin transform of their product is equal to the product of their Mellin transforms:

If s is restricted to integer values, a simpler result is

Thus the moments of the random product are the product of the corresponding moments of and this extends to non-integer moments, for example

.

The pdf of a function can be reconstructed from its moments using the Saddlepoint approximation method.

To illustrate how the product of moments yields a much simpler result than finding the moments of the distribution of the product, let be sampled from two Gamma distributions, with parameters whose moments are .

Multiplying the corresponding moments gives the Mellin Transform result


Independently, it is known that the product of two Gamma samples has the distribution .

To find the moments of this, make the change of variable , simplifying similar integrals to:

thus


The definite integral is well documented and we have finally


which, after some difficulty, has agreed with the moment product result above.

Special cases[edit]

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 convolved 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.

This works for the product of two independent variables each uniformly distributed on the x-interval [0,1]. Making the transformation , each is distributed on u as .
The convolution of the two distributions is the autoconvolution

Next retransform the variable to yielding the distribution
on the interval [0,1]

The product of two independent Normal samples follows a modified Bessel function. Let be samples from a Normal(0,1) distribution and . Then

The product of two independent Gamma samples, , defining , follows where

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.[4] 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.[4]

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).

In theoretical computer science[edit]

In computational learning theory, a product distribution over is specified by the parameters . Each parameter gives the marginal probability that the ith bit of sampled as is 1; i.e. . In this setting, the uniform distribution is simply a product distribution with every .

Product distributions are a key tool used for proving learnability results when the examples cannot be assumed to be uniformly sampled.[5] They give rise to an inner product on the space of real-valued functions on as follows:

This inner product gives rise to a corresponding norm as follows:

See also[edit]

Notes[edit]

  1. ^ Springer, Melvin Dale (1979). The Algebra of Random Variables. Wiley. ISBN 978-0-471-01406-5. Retrieved 24 September 2012.
  2. ^ Rohatgi, V. K. (1976). An Introduction to Probability Theory and Mathematical Statistics. Wiley Series in Probability and Statistics. New York: Wiley. doi:10.1002/9781118165676. ISBN 978-0-19-853185-2.
  3. ^ Grimmett, G. R.; Stirzaker, D.R. (2001). Probability and Random Processes. Oxford: Oxford University Press. ISBN 978-0-19-857222-0. Retrieved 4 October 2015.
  4. ^ a b Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1995). Continuous Univariate Distributions Volume 2, Second edition. Wiley. p. 306. ISBN 978-0-471-58494-0. Retrieved 24 September 2012.
  5. ^ Servedio, Rocco A. (2004), "On learning monotone DNF under product distributions", Inf. Comput., 193 (1): 57–74, doi:10.1016/j.ic.2004.04.003

References[edit]

  • Springer, Melvin Dale; Thompson, W. E. (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, Melvin Dale; Thompson, W. E. (1966). "The distribution of products of independent random variables". SIAM Journal on Applied Mathematics. 14 (3): 511–526. doi:10.1137/0114046. JSTOR 2946226.