A product distribution is a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
constructed as the distribution of the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s having two other known distributions. Given two
statistically independent
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...
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
The product is one type of algebra for random variables: Related to the product distribution are the
ratio distribution
A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions.
Given two (usually independent) random variables ''X'' ...
, sum distribution (see
List of convolutions of probability distributions
In probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability ...
) 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''.
Derivation for independent random variables
If
and
are two independent, continuous random variables, described by probability density functions
and
then the probability density function of
is
:
Proof
We first write the
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
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
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
and the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
. (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
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
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
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
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
.
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.
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
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
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
The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem, among other names, states that if X is a random variable whose expected v ...
:
:
In the inner expression, is a constant. Hence:
: