Page 45 - Probability and Statistical Inference

22 1. Notions of Probability

This F(x–) is the limit of F(h) as h converges to x from the left hand side of
x. In the Example 1.5.1, one can easily verify that F(1–) = 0, F(2–) = .2 and
F(4–) = .6. In other words, in this example, the jump of the df or the cdf F(x)
at the point

There is this natural correspondence between the jumps of a df, the left-
limit of a df, and the associated pmf. A similar analysis in the case of the Ex-
ample 1.5.2 is left out as the Exercise 1.5.4.

For a discrete random variable X, one can obtain P(X = x) as
F(x) – F(x–) where the left-limit F(x–) comes from (1.5.9).

Example 1.5.3 (Example 1.5.2 Continued) For the random variable X whose
pmf is given by (1.5.1), the probability that the total from the two dice will be
smaller than 4 or larger than 9 can be found as follows. Let us denote an event
A = {X < 4 ∪ X > 9} and we exploit (1.5.6) to write P(A) = P(X > 4) + P(X >
9) = P(X = 2, 3) + P(X = 10, 11, 12) = 9/36 = 1/4. What is the probability that
the total from the two dice will differ from 8 by at least 2? Let us denote an
event B = {|X – 8| ≥ 2} and we again exploit (1.5.6) to write P(B) = P(X ≤ 6) +
P(X ≥ 10) = P(X = 2, 3, 4, 5, 6) + P(X = 10, 11, 12) = 21/36 = 7/12. !
Example 1.5.4 (Example 1.5.3 Continued) Consider two events A, B de-
fined in the Example 1.5.3. One can obtain P(A), P(B) alternatively by using
the expression of the df F(x) given in the Example 1.5.2. Now, P(A) = P(X < 4)
+ P(X > 9) = F(3) + {1 – F(9)} = 3/36 + {1 – 30/36} = 9/36 = 1/4. Also, P(B)
= P(X ≤ 6) + P(X ≥ 10) = F(6) + {1 – F(9)} = 15/36 + {1 – 30/36} = 21/36 =
7/12. !
Next, let us summarize some of the important properties of a distribution
function defined by (1.5.5) associated with an arbitrary discrete random vari-
able.
Theorem 1.5.1 Consider the df F(x) with x ∈ ℜ, defined by (1.5.5), for a
discrete random variable X. Then, one has the following properties:
(i) F(x) is non-decreasing, that is F(x) ≤ F(y) for all x ≤ y where x, y ∈
ℜ;
(ii) ;

(iii) F(x) is right continuous, that is F(x + h) ↓ F(x) as h ↓ 0, for all x ∈ ℜ.