Sequence Comparison.pdf

B.3 Major Discrete Distributions 181
The Poisson distribution is probably the most important discrete distribution. It not
only has elegant mathematical properties but also is thought of as the law of the rare
events. For example, in a binomial distribution, if the number of trials n is large and
the probability of success p for each trial is small such that λ = np remains constant,
then the binomial distribution converges to the Poisson distribution with parameter
λ.
Applying integration by part, we see that the distribution function X of the Poisson
distribution with parameter λ is
F X (k)=Pr[X ≤ k]= λ k+1
k!
∫ ∞
1
y k e −λy dy, k = 0,1,2,....
(B.13)
B.3.5 Probability Generating Function
For a discrete random variable X, its probability generating function (pgf) is written
G(t) and defined as
G(t)=∑ Pr[X = x]t x ,
x∈I
(B.14)
where I is the set of all possible values of X. This sum function always converges to
1fort = 1 and often converges in an open interval containing 1 for all probability
distributions of interest to us. Probability generating functions have the following
two basic properties.
First, probability generating functions have one-to-one relationship with probability
distributions. Knowing the pgf is equivalent to knowing the probability distribution
for a random variable to certain extent. In particular, for a nonnegative
integer-valued random variable X, wehave
Pr[X = k]= 1 k!
d k G(t)
dt k ∣
∣∣∣t=0
. (B.15)
Second, if random variables X 1 ,X 2 ,...,X n are independent and have pgfs G 1 (t),
G 2 (t), ...,G n (t) respectively, then the pgf G(t) of their sum
is simply the product
X = X 1 + X 2 + ···+ X n
G(t)=G 1 (t)G 2 (t)···G n (t).
(B.16)