Page 59 - Probability and Statistical Inference

36 1. Notions of Probability

Recall the requirement in the part (ii) in (1.5.3) which demands that all the
probabilities given by (1.7.7) must add up to one. In order to verify this
directly, let us proceed as follows. We denote q = 1 – p and simply use the
expression for the sum of an infinite geometric series from (1.6.15), with m =
1, to write

The expression of the df for the Geometric (p) random variable is also quite
straight forward. One can easily verify that

Example 1.7.5 Some geological exploration may indicate that a well drilled
for oil in a region in Texas would strike oil with probability .3. Assuming that
the oil strikes are independent from one drill to another, what is the probability
that the first oil strike will occur on the sixth drill? Let X be the number of drills
until the first oil strike occurs. Then, X is distributed as Geometric (p = .3) so
–2.
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that one has P(X = 6) = (.3)(.7) ≈ 5.0421 × 10 !
Example 1.7.6 An urn contains six blue and four red marbles of identical
size and weight. We reach in to draw a marble at random, and if it is red, we
throw it back in the urn. Next, we reach in again to draw another marble at
random, and if it is red then it is thrown back in the urn. Then, we reach in for
the third draw and the process continues until we draw the first blue marble.
Let X be total number of required draws. Then, this random variable X has the
Geometric(p = .6) distribution. What is the probability that we will need to
draw marbles fewer than four times? Using the expression of the df F(x)
from (1.7.8) we immediately obtain P(X < 4) = F(3) = 1 – (.4) = .936.!
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The kind of sampling inspection referred to in the Example 1.7.6
falls under what is called sampling with replacement.

The Negative Binomial Distribution: A discrete random variable X is
said to have the negative binomial distribution with µ and k, customarily de-
noted by NB(µ, k), if and only if its pmf is given by

where 0 < μ, k < ∞. Here, µ and k are referred to as parameters. The
parameterization given in (1.7.9) is due to Anscombe (1949). This form of