Begin2.DVI

represent the binomial coefficients in the binomial expansion

(p+q)n=

(

n

0

)

pn+

(

n

1

)

pn−^1 q+

(

n

2

)

pn−^2 q^2 +···+

(

n

x

)

pxqn−x+··· +

(

n

n

)

qn= 1 (11.66)

In equations (11.62) and (11.66) the term

(n

x

)

represents the number of different

way of selecting x-objects from a collection of n-objects and the term

pxqn−x=pp︸ ︷︷···p︸

x times

qq︸ ︷︷···q︸

n-x times

(11 .67)

represents the probability of xsuccesses and n−xfailures in n-trials without regard to

any ordering of the arrangements of how the successes or failures occur. Consequently,

the equation (11.67) must be multiplied by the number of different arrangements

of the successes and failures and this is what produces the binomial probability

distribution.

The binomial distribution has the following properties

mean =μ=np and variance =σ^2 =npq (11 .68)

The figure 11-10 illustrates the binomial distribution for the parameter values n= 10

and p= 0. 2 , 0. 5 and 0. 9.

Figure 11-10. Selected binomial distributions for n= 10.

The Multinomial Distribution

The multinomial distribution occurs when many events can happen during a single

trial. If only one event can result from mmutually exclusive events E 1 , E 2 ,... , E m

occurring in a single trial, where p 1 , p 2 ,... , p m are the probabilities assigned to the

m-events, then the probability of getting n 1 E 1 ′s,n 2 E 2 ′s,... , n m E′msis given by the

multinomial probability function

f(n 1 , n 2 ,... , n m) =

n!

n 1 !n 2 !... n m!p

n 1

1 p

n 2

2 ···p

nm

m

where n 1 +n 2 +···+nm=nand p 1 +p 2 +···+pm= 1.