Page 216 - Elements of Distribution Theory

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202 Distribution Theory for Functions of Random Variables

Example 7.4 (Weibull distribution). Let X denote a random variable with a standard expo-
nential distribution so that X has an absolutely continuous distribution with distribution
function

F X (x) = 1 − exp(−x), x > 0

and density

p X (x) = exp(−x), x > 0.
1 1 θ
Let Y = X θ where θ> 0. The function g(x) = x θ has inverse x = y .It follows that
Y has an absolutely continuous distribution with distribution function
θ
F Y (y) = 1 − exp(−y ), y > 0
and density function

θ
p Y (y) = θy θ−1 exp(−y ), y > 0.
The distribution of Y is called a standard Weibull distribution with index θ.

7.3 Functions of a Random Vector

In this section, we consider the extension of Theorem 7.1 to the case of a random vector.
Let X denote a random vector with range X; consider a function g on X and let Y =
g(X). Because of the possible complexity of the function g,even when it is one-to-one,
an analogue of part (i) of Theorem 7.1 is not available. Part (ii) of Theorem 7.1, which
does not use the dimension of X in any meaningful way, is simply generalized to the vector
case. Part (iii) of the theorem, which is essentially the change-of-variable formula for
integration, is also easily generalized by using the change-of-variable formula for integrals
on a multidimensional space.
d
d
Recall that if g is a function from R to R , then the Jacobian matrix of g is the d × d
matrix with (i, j)th element given by ∂g i /∂x j where g i denotes the ith component of the
vector-valued function g; this matrix will be denoted by ∂g/∂x. The Jacobian of g at x is
the absolute value of the determinant of the Jacobian matrix at x, and is denoted by

.
∂g(x)
∂x

Theorem 7.2. Let X denote a d-dimensional random vector with an absolutely continuous
d
distribution with density function p X . Suppose that Y = g(X) where g : X → R denotes
a one-to-one continuously differentiable function. Let X 0 denote an open subset of X such
that Pr(X ∈ X 0 ) = 1 and such that the Jacobian of g is nonzero on X 0 . Then Y = g(X) has
an absolutely continuous distribution with density function p Y , given by

, y ∈ Y 0 ,
∂h(y)
∂y
p Y (y) = p X (h(y))
where Y 0 = g(X 0 ).