Distribution of the product of two random variables
probability distribution, in mathematics
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
Z
=
X
Y
{\displaystyle Z=XY}
is a product distribution.
The product distribution is the PDF of a product of two random variables. This is not the same as the product of the PDFs of two random variables. The concepts are sometimes ambiguously termed as in "product of Gaussians".
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.
Derivation for independent random variables
If
X
{\displaystyle X}
and
Y
{\displaystyle Y}
are two independent, continuous random variables, described by probability density functions
f
X
{\displaystyle f_{X}}
and
f
Y
{\displaystyle f_{Y}}
then the probability density function of
Z
=
X
Y
{\displaystyle Z=XY}
is
f
Z
(
z
)
=
∫
−
∞
∞
f
X
(
x
)
f
Y
(
z
/
x
)
1
|
x
|
d
x
.
{\displaystyle f_{Z}(z)=\int _{-\infty }^{\infty }f_{X}(x)f_{Y}(z/x){\frac {1}{|x|}}\,dx.}
Proof
We first write the cumulative distribution function of
Z
{\displaystyle Z}
starting with its definition
F
Z
(
z
)
=
def
P
(
Z
≤
z
)
=
P
(
X
Y
≤
z
)
=
P
(
X
Y
≤
z
,
X
≥
0
)
+
P
(
X
Y
≤
z
,
X
≤
0
)
=
P
(
Y
≤
z
/
X
,
X
≥
0
)
+
P
(
Y
≥
z
/
X
,
X
≤
0
)
=
∫
0
∞
f
X
(
x
)
∫
−
∞
z
/
x
f
Y
(
y
)
d
y
d
x
+
∫
−
∞
0
f
X
(
x
)
∫
z
/
x
∞
f
Y
(
y
)
d
y
d
x
{\displaystyle {\begin{aligned}F_{Z}(z)&\,{\stackrel {\text{def}}{=}}\ \mathbb {P} (Z\leq z)\\&=\mathbb {P} (XY\leq z)\\&=\mathbb {P} (XY\leq z,X\geq 0)+\mathbb {P} (XY\leq z,X\leq 0)\\&=\mathbb {P} (Y\leq z/X,X\geq 0)+\mathbb {P} (Y\geq z/X,X\leq 0)\\&=\int _{0}^{\infty }f_{X}(x)\int _{-\infty }^{z/x}f_{Y}(y)\,dy\,dx+\int _{-\infty }^{0}f_{X}(x)\int _{z/x}^{\infty }f_{Y}(y)\,dy\,dx\end{aligned}}}
We find the desired probability density function by taking the derivative of both sides with respect to
z
{\displaystyle z}
.
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