Distribution of the product of two random variables

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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 distribution is the PDF of the product of sample values. This is not the same as the product of their PDFs yet the concepts are often ambiguously termed as in "product of Gaussians".

Algebra of random variables

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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

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

 

Proof

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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.[3]

Alternate proof

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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  -  plane and an arc of constant   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.

Starting with  , we have  . So the probability increment is  . Since   implies  , we can relate the probability increment to the  -increment, namely  . Then integration over  , yields  .

A Bayesian interpretation

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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   have probability   at levels   with  . The conditional density is  . Therefore  .

Expectation of product of random variables

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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   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:

 
 

Variance of the product of independent random variables

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Let   be uncorrelated random variables with means   and variances  . If, additionally, the random variables   and   are uncorrelated, then the variance of the product XY is[4]

 

In the case of the product of more than two variables, if   are statistically independent then[5] the variance of their product is

 

Characteristic function of product of random variables

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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[6]

 

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

Mellin transform

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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.

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   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 independent Gamma-distributed samples (~Gamma(α,1) and Gamma(β,1)) has a K-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.

If X, Y are drawn independently from Gamma distributions with shape parameters   then

 

This type of result is universally true, since for bivariate independent variables   thus

 

or equivalently it is clear that   are independent variables.

Special cases

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Lognormal distributions

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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.

Uniformly distributed independent random variables

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Let   be the product of two independent variables   each uniformly distributed on the interval [0,1], possibly the outcome of a copula transformation. As noted in "Lognormal Distributions" above, PDF convolution operations in the Log domain correspond to the product of sample values in the original domain. Thus, making the transformation  , such that  , each variate is distributed independently on u as

 .

and the convolution of the two distributions is the autoconvolution

 

Next retransform the variable to   yielding the distribution

  on the interval [0,1]

For the product of multiple (> 2) independent samples the characteristic function route is favorable. If we define   then   above is a Gamma distribution of shape 1 and scale factor 1,   , and its known CF is  . Note that   so the Jacobian of the transformation is unity.

The convolution of   independent samples from   therefore has CF   which is known to be the CF of a Gamma distribution of shape  :

 .

Make the inverse transformation   to extract the PDF of the product of the n samples:

 

The following, more conventional, derivation from Stackexchange[7] is consistent with this result. First of all, letting   its CDF is

 

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 geometry of the product distribution of two random variables in the unit square.

The figure illustrates the nature of the integrals above. The area of the selection 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.

Independent central-normal distributions

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

 


The variance of this distribution could be determined, in principle, by a definite integral from Gradsheyn and Ryzhik,[8]

 

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 two Gaussian samples is often confused with the product of two Gaussian PDFs. The latter simply results in a bivariate Gaussian distribution.

Correlated central-normal distributions

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The product of correlated Normal samples case was recently addressed by Nadarajaha and Pogány.[9] Let   be zero mean, unit variance, normally distributed variates with correlation coefficient  

Then

 

Mean and variance: For the mean we have   from the definition of correlation coefficient. The variance can be found by transforming from two unit variance zero mean uncorrelated variables U, V. Let

 

Then X, Y are unit variance variables with correlation coefficient   and

 

Removing odd-power terms, whose expectations are obviously zero, we get

 

Since   we have

 

High correlation asymptote In the highly correlated case,   the product converges on the square of one sample. In this case the   asymptote is   and

 

which is a Chi-squared distribution with one degree of freedom.

Multiple correlated samples. Nadarajaha et al. further show that if   iid random variables sampled from   and   is their mean then

 

where W is the Whittaker function while  .

Using the identity  , see for example the DLMF compilation. eqn(13.13.9),[10] this expression can be somewhat simplified to

 

The pdf gives the marginal distribution of a sample bivariate normal covariance, a result also shown in the Wishart Distribution article. The approximate distribution of a correlation coefficient can be found via the Fisher transformation.

Multiple non-central correlated samples. The distribution of the product of correlated non-central normal samples was derived by Cui et al.[11] 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   denotes the double factorial.

If   are central correlated variables, the simplest bivariate case of the multivariate normal moment problem described by Kan,[12] then

 

where

  is the correlation coefficient and  

[needs checking]

Correlated non-central normal distributions

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The distribution of the product of non-central correlated normal samples was derived by Cui et al.[11] 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   are samples from a bivariate time series then the   is a Wishart matrix with K degrees of freedom. The product distributions above are the unconditional distribution of the aggregate of K > 1 samples of  .

Independent complex-valued central-normal distributions

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Let   be independent samples from a normal(0,1) distribution.
Setting   are independent zero-mean complex normal samples with circular symmetry. Their complex variances are  

The density functions of

  are Rayleigh distributions defined as:
 

The variable   is clearly Chi-squared with two degrees of freedom and has PDF

 

Wells et al.[13] show that the density function of   is

 

and the cumulative distribution function of   is

 

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   corresponds to the product of two independent Chi-square samples   each with two DoF. Writing these as scaled Gamma distributions   then, from the Gamma products below, the density of the product is

 

Independent complex-valued noncentral normal distributions

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The product of non-central independent complex Gaussians is described by O’Donoughue and Moura[14] and forms a double infinite series of modified Bessel functions of the first and second types.

Gamma distributions

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The product of two independent Gamma samples,  , defining  , follows[15]

 

Beta distributions

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Nagar et al.[16] define a correlated bivariate beta distribution

 

where

 

Then the pdf of Z = XY is given by

 

where   is the Gauss hypergeometric function defined by the Euler integral

 

Note that multivariate distributions are not generally unique, apart from the Gaussian case, and there may be alternatives.

Uniform and gamma distributions

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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.[17] 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.[17]

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).

Gamma and Pareto distributions

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The product of n Gamma and m Pareto independent samples was derived by Nadarajah.[18]

See also

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Notes

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  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. ^ Goodman, Leo A. (1960). "On the Exact Variance of Products". Journal of the American Statistical Association. 55 (292): 708–713. doi:10.2307/2281592. JSTOR 2281592.
  5. ^ Sarwate, Dilip (March 9, 2013). "Variance of product of multiple random variables". Stack Exchange.
  6. ^ "How to find characteristic function of product of random variables". Stack Exchange. January 3, 2013.
  7. ^ heropup (1 February 2014). "product distribution of two uniform distribution, what about 3 or more". Stack Exchange.
  8. ^ Gradsheyn, I S; Ryzhik, I M (1980). Tables of Integrals, Series and Products. Academic Press. pp. section 6.561.
  9. ^ Nadarajah, Saralees; Pogány, Tibor (2015). "On the distribution of the product of correlated normal random variables". Comptes Rendus de l'Académie des Sciences, Série I. 354 (2): 201–204. doi:10.1016/j.crma.2015.10.019.
  10. ^ Equ(13.18.9). "Digital Library of Mathematical Functions". NIST: National Institute of Standards and Technology.{{cite web}}: CS1 maint: numeric names: authors list (link)
  11. ^ a b Cui, Guolong (2016). "Exact Distribution for the Product of Two Correlated Gaussian Random Variables". IEEE Signal Processing Letters. 23 (11): 1662–1666. Bibcode:2016ISPL...23.1662C. doi:10.1109/LSP.2016.2614539. S2CID 15721509.
  12. ^ Kan, Raymond (2008). "From moments of sum to moments of product". Journal of Multivariate Analysis. 99 (3): 542–554. doi:10.1016/j.jmva.2007.01.013.
  13. ^ Wells, R T; Anderson, R L; Cell, J W (1962). "The Distribution of the Product of Two Central or Non-Central Chi-Square Variates". The Annals of Mathematical Statistics. 33 (3): 1016–1020. doi:10.1214/aoms/1177704469.
  14. ^ O’Donoughue, N; Moura, J M F (March 2012). "On the Product of Independent Complex Gaussians". IEEE Transactions on Signal Processing. 60 (3): 1050–1063. Bibcode:2012ITSP...60.1050O. doi:10.1109/TSP.2011.2177264. S2CID 1069298.
  15. ^ Wolfies (August 2017). "PDF of the product of two independent Gamma random variables". stackexchange.
  16. ^ Nagar, D K; Orozco-Castañeda, J M; Gupta, A K (2009). "Product and quotient of correlated beta variables". Applied Mathematics Letters. 22: 105–109. doi:10.1016/j.aml.2008.02.014.
  17. ^ 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.
  18. ^ Nadarajah, Saralees (June 2011). "Exact distribution of the product of n gamma and m Pareto random variables". Journal of Computational and Applied Mathematics. 235 (15): 4496–4512. doi:10.1016/j.cam.2011.04.018.

References

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  • 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.