Distributions of functions of two random variables
14.23 Distributions of functions of two random variables
We turn now to the following problem: If X and Y are one-dimensional random variables with known distributions, how do we find the distributionof new random variables such as X + Y, XY, or
This section describes a method that helps to answer questions like this. Two new random variables
are defined by equations of the form
where Y) or Y) is the particular combination in which we are interested. From
a knowledge of the joint distribution of the two-dimensional random variable (X, Y) we calculate the joint distribution
Once is known, the individual distribu- tions of
of
and V are easily found.
To describe the method in detail, we consider a one-to-one mapping of the xy-plane onto the
defined by the pair of equations
v=
Let the inverse mapping be given by
and assume that Q and R have continuous partial derivatives. If T denotes a region in the xy-plane, let T’ denote its image in the uv-plane, as suggested by Figure 14.14. Let X and
Y be two one-dimensional continuous random variables having a continuous joint distribu- tion and assume (X, Y) has a probability density
Define new random variables and V by writing
Y). To determine a probability density of the random variable (U, V) we proceed as follows: The random variables X and Y are associated with a sample space S. For each
in S we have
Since the mapping is one-to-one, the two sets
Y(w)] and V(w ) =
and
are equal. Therefore we have (14.37)
V)
the density function of (X, Y) we can write
Using (14.37) and the formula for transforming a double integral we rewrite (14.38) as follows :
Y) T'] =
du dv.
I,I
Distributions of functions of two random variables
F IGURE 14.14 A one-to-one mapping of a region Tin the xy-plane onto a region in
the uv-plane.
Since this is valid for every region T’ in the uu-plane a density of is given by the integrand on the right; that is, we have
The densities and can now be obtained by the integration formulas
of two random variables. Given two one-dimensional random variables
EXAMPLE 1. The sum and
and Y with joint density determine density functions for the random variables
Yand
X- Y.
We use the mapping given by = x + , v = . This is a nonsingular linear transformation whose inverse is given by
The Jacobian determinant is
Calculus
of probabilities
Applying Equation (14.39) we see that a joint density g of is given by the formula
To obtain a density
we integrate with respect to and find
The change of variable x =
dx = du, transforms this to
Similarly, we find
An important special case occurs when and Y are independent. In this case the joint probability density factors into a product,
and the integrals for
and
become
EXAMPLE 2. The sum of two exponential distributions. Suppose now that each of and Y has an exponential distribution, say
= = 0 for < 0, and
for
Determine the density of
+ Y when
and Y are independent.
If < 0 the integral for is 0 since the factor = 0 for x < 0, and the factor
Solution.
x) = 0 for x 0. If 0 the integral for
becomes
dx =
dx .
To evaluate the last integral we consider two cases, = and 1. If = the integral has the value and we obtain
for
If we obtain for
Exercises
3. The maximum and minimum of two independent random variables. Let and Y be two independent one-dimensional random variables with densities
EX AM PLE
and
and
corresponding distribution functions
be the random variables
Y},
Y}.
That is, for each in the sample space,
is the minimum of the two numbers X(o), Y(o). The mapping = max
is the maximum and
v = min {x, is not to-one, so the procedure used to deduce Equation (14.39) is not applicable. However, in this case we can obtain the distribution functions of
directly from first principles. First we note that
and
. Therefore t) = Y
if, and only if,
and Y
By independence this is equal to t)P( Y t) = Thus, we have
At each point of continuity
and
we can differentiate this relation to obtain
Similarly, we have > if and only if > and Y > Therefore =
> t)P( Y > t)
At points of continuity of
and
we differentiate this relation to obtain