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372 Appendix A Random Variables and Probability Distributions
We shall see in Example A.1.2 below that λ is the mean of X.
(i) The negative binomial distribution with parameters α and p. The random variable
X is said to have a negative binomial distribution with parameters α>0 and
p ∈ [0, 1] if it has pmf
j
k − 1 + α
nb(j; α, p)
(1 − p)
k
j p α , j 0, 1,...,
k1
where the product is defined to be 1 if j 0.
Not all random variables can be neatly categorized as either continuous or discrete.
For example, consider the time you spend waiting to be served at a checkout
counter and suppose that the probability of finding no customers ahead of you is 1
2 .
Then the time you spend waiting for service can be expressed as
⎧
⎪⎨ 0, with probability
W
⎪⎩
1
2 ,
W1, with probability 1
2 ,
where W1 is a continuous random variable. If the distribution of W1 is exponential
with parameter 1, then the distribution function of W is
⎧
⎨ 0, if x0). It is expressible as a mixture,
F pFd + (1 − p)Fc,
with p 1,
of a discrete distribution function
2
0, x < 0,
Fd
1, x ≥ 0,
and a continuous distribution function
0, x < 0,
Fc
1 − e −x , x ≥ 0.
Every distribution function can in fact be expressed in the form
F p1Fd + p2Fc + p3Fsc,
where 0 ≤ p1,p2,p3 ≤ 1, p1 + p2 + p3 1, Fd is discrete, Fc is continuous, and Fsc
is singular continuous (continuous but not of the form A.1.2). Distribution functions
with a singular continuous component are rarely encountered.