Sequence Comparison.pdf

184 B Elementary Probability Theory
Table B.1 Means and variances of major probability distributions.
Distribution Parameters Mean Variance
Bernoulli p p p(1 − p)
Binomial p,n np np(1 − p)
Geometric p p/1 − p p/(1 − p) 2
Poisson λ λ λ
Uniform [a,b] (a + b)/2 (b − a) 2 /12
Exponential λ 1/λ 1/λ 2
Normal µ, σ 2 µ σ 2
µ X = dG(t)
dt
∣
∣
t=1
(B.23)
If X is a discrete random variable with range I and g is a function on I, then g(X)
is also a discrete random variable. The expected value of g(X) is written E(g(X))
and defined as
E(g(X)) = ∑ g(x)Pr[X = x].
x∈I
(B.24)
Similarly, for a continuous random variable Y having density function f Y , its
mean is defined as
∫ ∞
µ Y = yf Y (y)d(y).
(B.25)
−∞
If g is a function, the expected value of g(Y ) is defined as
∫ ∞
E(g(Y )) = g(y) f Y (y)d(y).
−∞
(B.26)
The means of random variables with a distribution described in Sections B.3 and
B.4 are listed in Table B.1.
Let X 1 ,X 2 ,...,X n be random variables having the same range I. Then, for any
constants a 1 ,a 2 ,...,a n ,
E(
n
∑
i=1
a i X i )=
n
∑
i=1
a i E(X i ).
(B.27)
This is called the linearity property of the mean.
For any positive integer, E(X m ) is called the rth moment of a random variable X
provided that the sum exists.