Page 48 - Probability and Statistical Inference

1. Notions of Probability 25

Theorem 1.6.1 Suppose that F(x), x ∈ ℜ, is the df of an arbitrary random
variable X. Then, the set of points of discontinuity of the distribution function
F(x) is finite or at the most countably infinite.
Example 1.6.1 Consider the following discrete random variable X first:

In this case, the df F (x) is discontinuous at the finite number of points x = –
X
1, 2 and 5. !
Example 1.6.2 Look at the next random variable Y. Suppose that

In this case, the corresponding df F (y) is discontinuous at the countably
Y
infinite number of points y = 1, 2, 3, ... .
Example 1.6.3 Suppose that a random variable U has an associated non-
negative function f(u) given by

Observe that and thus f(u) happens to
be the distribution of U. This random variable U is neither discrete nor con-
tinuous. The reader should check that its df F (u) is discontinuous at the count-
U
ably infinite number of points u = 1, 2, 3, ... . !
If a random variable X is continuous, then its df F(x) turns out to be continu-
ous at every point x ∈ ℜ. However, we do not mean to imply that the df F(x)
will necessarily be differentiable at all the points. The df F(x) may not be differ-
entiable at a number of points. Consider the next two examples.
Example 1.6.4 Suppose that we consider a continuous random variable W
with its pdf given by

In this case, the associated df F (w) is given by
W