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A '''product distribution''' is a [[probability distribution#Continuous case|probability distribution]] constructed as the distribution of the [[Product (mathematics)|product]] of [[random variable]]s 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
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: <math>Z = XY</math>
 
is a ''product distribution''.
 
== Algebra of random variables ==
{{main|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''.<ref name="SpringerM1979Algebra">{{Cite book
| last = Springer
| first = Melvin Dale
| title = The Algebra of Random Variables
| url = http://books.google.com/books/about/The_algebra_of_random_variables.html?id=qUDvAAAAMAAJ
| accessdate = 24 September 2012
| publisher = [[John Wiley & Sons|Wiley]]
| year = 1979
| isbn = 0-471-01406-0
}}</ref>
 
== Derivation ==
If <math>X </math> and <math>Y </math> are two independent, continuous random variables, described by probability density functions <math>f_X </math> and <math>f_Y </math> then  the [[Joint_distribution#Continuous_case|joint distribution]] of <math>Z = XY</math> is<ref>{{cite book|last=Rohatgi|first=V. K.|title=An Introduction to Probability Theory and Mathematical Statistics|year=1976|publisher=Wiley|location=New York|isbn=978-0471731351|url=http://books.google.com/books?id=YyXvAAAAMAAJ&dq=editions:UOM39015026067572|accessdate=24 September 2012}}</ref>
 
:<math>f_Z(z) = \int^{\infty}_{-\infty} f_X \left( x \right)  f_Y \left( z/x \right)  \frac{1}{|x|}\, dx, </math>
 
and when both <math>X </math> and <math>Y </math> are described by the same probability density function <math>f </math> then
 
:<math>f_Z(z) = \int^\infty_{-\infty} f \left( x\right) f \left( z/x \right)  \frac{1}{|x|} \, dx. </math>
 
==Special cases==
The distribution of the product of two random variables which have [[lognormal distribution]]s 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.
 
The distribution of the product of a random variable having a [[Uniform distribution (continuous)|uniform distribution]] on (0,1) with a random variable having a [[gamma distribution]] with shape parameter equal to 2, is an [[exponential distribution]].<ref name=JKB>
{{cite book
|last1=Johnson |first1=Norman L.
|last2=Kotz |first2=Samuel
|last3=Balakrishnan |first3= N.
|title=Continuous Univariate Distributions Volume 2, Second edition
|url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471584940.html
|accessdate=24 September 2012
|page=306
|year=1995
|publisher=Wiley
|isbn=0-471-58494-0
}}</ref> 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.<ref name=JKB />
 
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==
 
In [[computational learning theory]], a '''product distribution''' <math>\mathcal{D}</math> over <math>\{0,1\}^n</math> is specified by the parameters
<math>\mu_1, \mu_2, \dots, \mu_n</math>. Each parameter <math>\mu_i</math> gives the marginal probability that the ''i''<sup>th</sup> bit of
<math>x \in \{0,1\}^n</math> sampled as <math>x \sim \mathcal{D}</math> is 1; i.e.
<math>\mu_i = \operatorname{Pr}_{\mathcal{D}}[x_i = 1]</math>. In this setting, the uniform distribution is simply a product distribution with every <math>\mu_i = 1/2</math>.
 
Product distributions are a key tool used for proving learnability results when the examples cannot be assumed to be uniformly sampled.<ref>
{{Citation
| last1 = Servedio | first1 = Rocco A.
| title = On learning monotone DNF under product distributions
| journal = Inf. Comput.
| volume = 193
| issue = 1
| pages = 57–74
| publisher = Academic Press, Inc.
| year = 2004
| doi = 10.1016/j.ic.2004.04.003}}
</ref> They give rise to an [[inner product]]
<math>\langle \cdot, \cdot \rangle</math> on the space of real-valued functions on <math>\{0,1\}^n</math> as follows:
 
: <math>
\langle f, g \rangle_\mathcal{D} = \sum_{x \in \{0,1\}^n} \mathcal{D}(x)f(x)g(x) = \mathbb{E}_{\mathcal{D}} [fg]
</math>
 
This inner product gives rise to a corresponding [[Norm (mathematics)|norm]] as follows:
 
: <math>
\|f\|_{\mathcal{D}} = \sqrt{ \langle f, f \rangle_\mathcal{D} }
</math>
 
==See also==
* [[Algebra of random variables]]
 
==Notes==
{{Reflist}}
 
==References==
*{{cite journal
  | doi = 10.1137/0118065
  | title = The distribution of products of beta, gamma and Gaussian random variables
  | last1 = Springer
  | first1 = Melvin Dale
  | last2 = Thompson
  | first2 = W. E.
  | journal = SIAM Journal on Applied Mathematics
  | volume = 18
  | issue = 4
  | pages = 721&ndash;737
  | year = 1970
  | jstor = 2099424
}}
*{{cite journal
  | title = The distribution of products of independent random variables
  | last1 = Springer
  | first1 = Melvin Dale
  | last2 = Thompson
  | first2 = W. E.
  | journal = SIAM Journal on Applied Mathematics
  | pages = 511&ndash;526
  | year = 1966
  | jstor = 2946226
  | volume = 14
  | issue = 3
  | doi = 10.1137/0114046
}}
 
{{DEFAULTSORT:Product Distribution}}
[[Category:Types of probability distributions]]
[[Category:Algebra of random variables]]

Latest revision as of 21:16, 5 May 2014

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