Page 216 - Probability and Statistical Inference

4. Functions of Random Variables and Sampling Distribution 193

Y = g(X). Suppose that the domain space of Y is also a subinterval of ℜ.
Then, the pdf of Y is given by

for y ∈ γ.
That is, in the expression of f(x), the x variable is first replaced by the new
variable g (y) everywhere and then we multiply the resulting expression by the abso-
–1
lute value of the Jacobian of transformation which is Incidentally, note
that is evaluated as the absolute value of dx/dy and then replacing
the variable x ∈ χ in terms of the new variable Y ∈ .

Figure 4.4.1. A Mapping of x to y

Recall the convention that when a pdf is written down with its
support, it is understood that the densitY is zero elsewhere.

Example 4.4.1 (Example 4.2.5 Continued) We write W = F(X) where we
have g(x) = F(x) for x ∈ (a, b), so that x = F (w) for w for w ∈ (0, 1). This
–1
transformation from x to w is one-to-one. Now, dw/dx = f(x) = f(F (w)).
–1
Then, using (4.4.1) the pdf of W becomes

In other words, W = F(X) is distributed uniformly on the interval (0, 1). Next,
we have U = q(W) where q(w) = –log(w) for 0 < w < 1. This transformation
from w to u is also one-to-one. Using (4.4.1) again, the pdf of U is obtained as
follows:

which shows that U is distributed as a standard exponential variable or
Gamma(1, 1). !