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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 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]
If
We first write the cumulative distribution function of
We find the desired probability density function by taking the derivative of both sides with respect to
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
where
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
Starting with
Let
Letting
For the case of one variable being discrete, let
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 in which case
Let
In the case of the product of more than two variables, if
Assume X, Y are independent random variables. The characteristic function of X is
If the characteristic functions and distributions of both X and Y are known, then alternatively,
The Mellin transform of a distribution
The inverse transform is
if
If s is restricted to integer values, a simpler result is
Thus the moments of the random product
The pdf of a function can be reconstructed from its moments using the saddlepoint approximation method.
A further result is that for independent X, Y
Gamma distribution example To illustrate how the product of moments yields a much simpler result than finding the moments of the distribution of the product, let
Multiplying the corresponding moments gives the Mellin transform result
Independently, it is known that the product of two independent Gamma samples has the distribution
To find the moments of this, make the change of variable
thus
The definite integral
which, after some difficulty, has agreed with the moment product result above.
If X, Y are drawn independently from Gamma distributions with shape parameters
This type of result is universally true, since for bivariate independent variables
or equivalently it is clear that
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.
Let
and the convolution of the two distributions is the autoconvolution
Next retransform the variable to
For the product of multiple ( >2 ) independent samples the characteristic function route is favorable. If we define
The convolution of
Making the inverse transformation
The following, more conventional, derivation from Stackexchange[6] is consistent with this result. First of all, letting
The density of
Multiplying by a third independent sample gives distribution function
Taking the derivative yields
The author of the note conjectures that, in general,
The figure illustrates the nature of the integrals above. The shaded area within the unit square and below the line z = xy, represents the CDF of z. This divides into two parts. The first is for 0 < x < z where the increment of area in the vertical slot is just equal to dx. The second part lies below the xy line, has y-height z/x, and incremental area dx z/x.
The product of two independent Normal samples follows a modified Bessel function. Let
The variance of this distribution could be determined, in principle, by a definite integral from Gradsheyn and Ryzhik,[7]
thus
A much simpler result, stated in a section above, is that the variance of the product of zero-mean independent samples is equal to the product of their variances. Since the variance of each Normal sample is one, the variance of the product is also one.
The product of correlated Normal samples case was recently addressed by Nadarajaha and Pogány.[8] Let
Then
Mean and variance: For the mean we have
Then X, Y are unit variance variables with correlation coefficient
Removing odd-power terms, whose expectations are obviously zero, we get
Since
High correlation asymptote In the highly correlated case,
which is a Chi-squared distribution with one degree of freedom.
Multiple correlated samples. Nadarajaha et. al. further show that if
where W is the Whittaker function while
Using the identity
The pdf gives the distribution of a sample covariance.
Multiple non-central correlated samples. The distribution of the product of correlated non-central normal samples was derived by Cui et.al.[10] and takes the form of an infinite series of modified Bessel functions of the first kind.
Moments of product of correlated central normal samples
For a central normal distribution N(0,1) the moments are
where
If
where
[needs checking]
The distribution of the product of non-central correlated normal samples was derived by Cui et al.[10] and takes the form of an infinite series.
These product distributions are somewhat comparable to the Wishart distribution. The latter is the joint distribution of the four elements (actually only three independent elements) of a sample covariance matrix. If
Let
Setting
The density functions of
The variable
Wells et. al.[12] show that the density function of
and the cumulative distribution function of
Thus the polar representation of the product of two uncorrelated complex Gaussian samples is
The first and second moments of this distribution can be found from the integral in Normal Distributions above
Thus its variance is
Further, the density of
The product of non-central independent complex Gaussians is described by O’Donoughue and Moura[13] and forms a double infinite series of modified Bessel functions of the first and second types.
The product of two independent Gamma samples,
Nagar et. al.[15] define a correlated bivariate beta distribution
where
Then the pdf of Z = XY is given by
where
Note that multivariate distributions are not generally unique, apart from the Gaussian case, and there may be alternatives.
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.[16] 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.[16]
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).
The product of n Gamma and m Pareto independent samples was derived by Nadarajah.[17]
In computational learning theory, a product distribution
Product distributions are a key tool used for proving learnability results when the examples cannot be assumed to be uniformly sampled.[18] They give rise to an inner product
This inner product gives rise to a corresponding norm as follows: