Geoscience Reference
In-Depth Information
A.7.1.2 Binomial Distribution
The binomial distribution is the distribution of the sum of a number of Bernoulli
variables with the same probability of the value 1. Two parameters determine a
binomial distribution: the probability q of 1 in each of the Bernoulli summands and
the number n of summands.
The classical example of a random variable with binomial distribution is the
number of successes in a number of independent trials each of them with the same
probability. Let S be the set of possible sequences of results of tossing n times a
coin that has the probability q of showing the side with a head and let X be the
number of heads observed. X has a binomial distribution with parameters n and q
(X
Binomial(n,q)).
Since the expected value of a sum of random variables is the sum of the expected
values of the summands,
*
EX ¼ nq :
By independence,
VX
ðÞ ¼
nq 1
ð
q
Þ:
Employing independence and a simple combinatorial computation, it may also
be proved that, for every integer k from 0 to n,
Þ n k :
q k 1
ð
pX
½
¼
k
¼
f
n
! =
½
k
!
ð
n
k
Þ !
g
q
From this and Newton Binomial Formula for the power of a sum follows
X n
p
½
X
¼
k
¼
1
:
k¼0
A.7.1.3 Poisson Distribution
X has a Poisson distribution in the sample space N of the natural numbers, with
parameter
ʻ
(X
Poisson(
ʻ
)) for a positive real
ʻ $
*
e k k k =
pX
½
¼
k
¼
k
! ;
for every k
2
N
:
An example of random variable with Poisson distribution is given by the number
of particles emitted by a radioactive source in a given time.
Notice that P k¼0 k k =
k! = e k , so that, in fact, if X has a Poisson distribution,
then
X n
p
½
X
¼
k
¼
1
:
k ¼ 0
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