Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

For evaluating binomial probabilities a useful result is the binomial recurrence

formula

Pr(X=x+1)=

p

q

(

n−x

x+1

)

Pr(X=x), (30.95)

which enables successive probabilities Pr(X=x+k),k=1, 2 ,..., to be calculated

once Pr(X=x) is known; it is often quicker to use than (30.94).

The random variableXis distributed asX∼Bin(3,^12 ). Evaluate the probability function

f(x)using the binomial recurrence formula.

The probability Pr(X= 0) may be calculated using (30.94) and is

Pr(X=0)=^3 C 0

( 1

2

) 0 ( 1

2

) 3

=^18.

The ratiop/q=^12 /^12 = 1 in this case and so, using the binomial recurrence formula
(30.95), we find

Pr(X=1)=1×

3 − 0

0+1

×

1

8

=

3

8

,

Pr(X=2)=1×

3 − 1

1+1

×

3

8

=

3

8

,

Pr(X=3)=1×

3 − 2

2+1

×

3

8

=

1

8

,

results which may be verified by direct application of (30.94).

We note that, as required, the binomial distribution satifies

∑n

x=0

f(x)=

∑n

x=0

nC

xp

xqn−x=(p+q)n=1.

Furthermore, from the definitions ofE[X]andV[X] for a discrete distribution,

we may show that for the binomial distributionE[X]=npandV[X]=npq.The

direct summations involved are, however, rather cumbersome and these results

are obtained much more simply using the moment generating function.

The moment generating function for the binomial distribution

To find the MGF for the binomial distribution we consider the binomial random

variableXto be the sum of the random variablesXi,i=1, 2 ,...,n, which are

defined by

Xi=

{

1 if a ‘success’ occurs on theith trial,

0 if a ‘failure’ occurs on theith trial.