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Important discrete distributions

Table 1.1 – Some important discrete probability distributions

Distribution Probability law f(x) E(X) Var(X)
x x n−x
binomial C p q np npq
n
x
r x
negative binomial C r+x−1 p q rq rq 2
p
p
geometric q x−1 p 1 q
p p 2
hypergeometric (Np)!(Nq)!n!(N−n)! np N−n npq
x!(Np−x)!(n−x)!(Nq−n+x)!N! N−1
Poisson λ x e −λ λ λ
x!
It is convenient to introduce the notation ν k for the k-th central moment. Thus Var(X) ≡ ν 2 and
2
we may write (5.10) as ν 2 = µ 2 − µ . Clearly, the first central moment of a distribution is always
1
zero since.
We note that the notation µ k and ν k for the moments and central moments respectively is not
universal. Indeed, in some books their meanings are reversed. We can write the k-th central
k
moment of a distribution in terms of its k-th and lower-order moments by expanding (X − µ) in
2
powers of X. We have already noted that ν 2 = µ 2 − µ , and similar expressions may be obtained
1
for higher-order central moments. For example,

2
3
3
2
3
2
3
3
ν 3 = E(X−µ 1 ) = E(X −3µ 1 X +3mu X−µ ) = µ 3 −3µ 1 µ 2 +3µ µ 1 −µ == µ 3 −3µ 1 µ 2 +2µ .
1 1 1 1 1
(5.12)
In general, it is straightforward to show that
r
1
r
r
k
ν k = µ k − C µ k−1 µ 1 + · · · + (−1) C µ k−r µ + · · · + (−1) k−1 (C k−1 − 1)µ . (5.13)
k k 1 k 1
One may also characterize a probability distribution f(x) using the closely related normalized
and dimensionless central moments
ν k ν k
γ k = = .
ν 2 k/2 σ k
From this set, γ 3 and γ 4 are more commonly called, respectively, the skewness and kurtosis of
the distribution. The skewness γ 3 of a distribution is zero if it is symmetrical about its mean. If
the distribution is skewed to values of x smaller than the mean then γ 3 < 0. Similarly γ 3 > 0 if the
distribution is skewed to higher values of x.

Important discrete distributions

Having discussed the some general properties of distributions, we now consider the more
important discrete distributions encountered in technical applications.
These are discussed in detail below, and summarized for convenience in table 1.1; we refer the
reader to the relevant section below for an explanation of the symbols used.

Binomial distribution. Let p be the probability that an event will happen in any single Bernoulli
trial (called the probability of success). Then q = 1 − p is the probability that the event will fail
to happen in any single trial (called the probability of failure). The probability that the event will
happen exactly x times in n trials (i.e. x successes and n − x failures will occur) is given by the
probability function
x x n−x
f(x) = P(X = x) = C p q , (5.14)
n
where the random variable X denotes the number of successes in n trials and x = 0, 1, . . . , n.

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