Sequence Comparison.pdf

180 B Elementary Probability Theory
B.3.2 Binomial Distribution
A binomial random variable X is the number of successes in a fixed number n of
independent Bernoulli trials with parameter p.Thep and n are called the parameter
and index of X, respectively. In particular, the Bernoulli trial can be considered as a
binomial distribution with index 1.
The probability distribution of the binomial random variable X with index n and
parameter p is
Pr[X = k]=
n!
k!(n − k)! pk (1 − p) n−k , k = 0,1,...,n.
(B.9)
B.3.3 Geometric and Geometric-like Distributions
A random variable X has a geometric distribution with parameter p if
Pr[X = k]=(1 − p)p k , k = 0,1,2,....
(B.10)
The geometric distribution arises from independent Bernoulli trials. Suppose that a
sequence of independent Bernoulli trials are conducted, each trial having probability
p of success. The number of successes prior to the first failure has the geometric
distribution with parameter p.
By (B.10), the distribution function of the geometric random variable X with
parameter p is
F X (x)=1 − p x+1 , x = 0,1,2,....
(B.11)
Suppose that Y is a random variable taking possible values 0, 1, 2, ...and 0<
p < 1. If
1 − F Y (y)
lim
y→∞ p y+1 = C
for some fixed C, then the random variable Y is said to be geometric-like. Geometriclike
distributions play a central role in BLAST statistic theory studied in Chapter 7.
B.3.4 The Poisson Distribution
A random variable X has the Poisson distribution with parameter λ if
Pr[X = x]= 1 x! e−λ λ x , x = 0,1,2,....
(B.12)