Page 60 - Probability and Statistical Inference

1. Notions of Probability 37

the pmf is widely used in the areas such as entomology, plant science,
and soil science. The parameters μ and k have physical interpretations in
many applications in these areas.
Denoting p = k/(µ + k), q = µ/(µ + k), the pmf given by (1.7.9) can be
rewritten in a more traditional way as follows:

where 0 < p < 1, q = 1 – p and k is a positive integer. This form of the
negative binomial distribution arises as follows. Suppose that we have the
same basic setup as in the case of a geometric distribution, but instead we let
th
X be the number of 0’s observed before the k occurrence of 1. Then, we
have P(X = 0) = P(Observing k many 1’s right away) = p . Next, P(X =
k
P(The last trial yields 1, but we observe k – 1 many 1’s and a single 0 before
the occurrence of the 1 in the last trial) = . Also,
P(X = 2) = P(The last trial yields 1, but we observe k – 1 many 1’s and two
0’s before the occurrence of the 1 in the last trial) =
. Analogous arguments will eventually justify
(1.7.10) in general.
Example 1.7.7 (Example 1.7.5 Continued) Some geological exploration
indicates that a well drilled for oil in a region in Texas may strike oil with
probability .3. Assuming that the oil strikes are independent from one drill to
another, what is the probability that the third oil strike will occur on the tenth
well drilled? Let X be the number of drilled wells until the third oil strike
occurs. Then, using (1.7.10) we immediately get P(X = 10) = (.7) (.3) ≈
3
7
–2
8.0048 × 10 . !
The Discrete Uniform Distribution: Let X be a discrete random variable
which takes the only possible values x , ..., x each with the same probability
k
1
1/k. Such X is said to have a discrete uniform distribution. We may write
down its pmf as follows.

Example 1.7.8 Suppose that we roll a fair die once and let X be the num-
ber of dots on the face of the die which lands up. Then, obviously f(x) = 1/6
for x = 1, ..., 6 which corresponds to the pmf given by (1.7.11) with k = 6
and x = 1, x = 2, ..., x . !
1 2 6
1.7.2 Continuous Distributions
In this subsection we include some standard continuous distributions. A few
of these appear repeatedly throughtout the text.