Sequence Comparison.pdf

B.3 Major Discrete Distributions 179
A random variable is discrete if it takes a value from a discrete set of numbers.
For example, the number of heads turning up in the experiment of tossing a coin n
times is a discrete random variable. In this case, the possible values of the random
variableare0,1,2,...,n.Theprobability distribution of a random variable is often
presented in the form of a table, a chart, or a mathematical function. The distribution
function of a discrete random variable X is written F X (x) and defined as
F X (x)=∑ Pr[X = v], −∞ < x < ∞.
v≤x
(B.6)
Notice that F X is a step function for a discrete random variable X.
A random variable is continuous if it takes any value from a continuous interval.
The probability for a continuous random variable is not allocated to specific values,
but rather to intervals of values. If there is a nonnegative function f (x) defined for
−∞ < x < ∞ such that
∫ b
Pr[a < X ≤ b]= f (x)dx, ∞ < a < b < ∞,
a
then f (x) is called the probability density function of the random variable X. IfX
has a probability density function f X (x), then its distribution function F X (x) can be
written as
∫ x
F X (x)=Pr[X ≤ x]= f X (x), −∞ < x < ∞.
(B.7)
−∞
B.3 Major Discrete Distributions
In this section, we simply list the important discrete probability distributions that
appear frequently in bioinformatics.
B.3.1 Bernoulli Distribution
A Bernoulli trial is a single experiment with two possible outcomes “success” and
“failure.” The Bernoulli random variable X associated with a Bernoulli trial takes
only two possible values 0 and 1 with the following probability distribution:
Pr[X = x]=p x (1 − p) 1−x , x = 0,1,
(B.8)
where p is called the parameter of X.
In probability theory, Bernoulli random variables occur often as indicators of
events. The indicator I A of an event A is the random variable that has 1 if A occurs
and 0 otherwise. The I A is a Bernoulli random variable with parameter Pr[A].