Page 50 - Probability and Statistical Inference

1. Notions of Probability 27

At all points x ∈ ℜ wherever the df F(x) is differentiable, one must have
the following result:

This is a simple restatement of the fundamental theorem of integral calculus.
Example 1.6.7 (Example 1.6.4 Continued) Consider the df F (w) of the
W
random variable W from (1.6.8). Now, F (w) is not differentiable at the two
W
points w = 1, 2. Except at the points w = 1, 2, the pdf f(w) of the random
variable W can be obtained from (1.6.8) as follows: f(w) = d/dw F (w) which
W
will coincide with zero when –∞ < w < 1 or 2 < w < ∞, whereas for 1 < w < 2
we would have d{1/7(w – 1)}/dw = 3/7w . This agrees with the pdf given by
2
3
(1.6.7) except at the points w = 1, 2. !

For a continuous random variable X with its pdf f(x and
its df F(x), x ∈ ℜ, one has: (i) P(X < a) or P(X ≤ a) is
given by F(a) = , (ii) P(X > b) or P(X ≥ b) is
given by 1-F(b) = , and (iii) P(a < X < b) or
P(a ≤ X < b) or P (a < X ≤ b) or P(a ≤ X ≤ b) is given
by F(b) - F(a) = . The general understanding,
however, is that these integrals are carried out within the
appropriate intervals wherever the pdf f(x) is positive.

In the case of the Example 1.6.7, notice that the equation (1.6.10) does not
quite lead to any specific expression for f(w) at w = 1, 2, the points where F (w)
W
happens to be non-differentiable.. So, must f(w) be defined at w = 1, 2 in exactly
the same way as in (1.6.7)?
Let us write I(.) for the indicator function of (.). Since we only handle
integrals when evaluating probabilities, without any loss of generality, the pdf
given by (1.6.7) is considered equivalent to any of the pdf’s such as 3/7w I(1
2
2
2
≤ w < 2) or 3/7w I(1 < w ≤ 2) or 3/7w I(1 ≤ w ≤ 2). If we replace the pdf f(w)
from (1.6.7) by any of these other pdf’s, there will be no substantive changes
in the probability calculations.
From this point onward, when we define the various pieces of the
pdf f(x) for a continuous random variable X, we will not attach
any importance on the locations of the equality signs placed to
identify the boundaries of the pieces in the domain of the variable x.