Probability Distributions - Oxford University Press

226 Business Statistics Using Excel
❉ Interpretation We can see from Excel that the binomial and Poisson distributions
provide approximately equal results, 45.92% and 46.28% respectively.
The degree of agreement between the binomial and Poisson probability distributions
for this problem can be observed in Figure 5.43. They virtually overlap.
P(X = r)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Comparison between binomial and Poisson
distributions
0 5 10
X
Note The solution process is as follows:
15
Binomial
Poisson
Figure 5.43
1. Binomial Solution: P(less than 2 rotten) = P(X < 2) = P(X = 0) + P(X = 1).
− −
( ) 0
1
0 60
1 59
0 ( ) ( ) 1(
) ( )
60 0 600 60 1 601 60
60
PX< 2 = C pq + Cpq = C 0.3 0.97 + C 0.3 0.97
P( X< 2)=
0. 1608 + 0. 2984 = 0. 4592
2. Poisson Solution: P(less than 2 rotten) = P(X < 2) = P(X = 0) + P(X = 1). Since n is large and
p is small we can use the Poisson distribution. To check we will see if the mean and the
variance of the distribution are equal: Mean = np = 60 * 0.03 = 1.8, Variance = npq = 60 *
0.03 * 0.97 = 1.746. Comparing the two values we see that they are approximately equal
and the binomial distribution can be approximated using the Poisson distribution:
0 −1.8 1 −1.8
1.8 e 1.8e
P( X< 2)
= + = 0. 1652 + 0. 2975=
0. 4627
0! 1!
Table 5.15 compares the two solutions:
Problem Binomial Poisson
P(X < 2) 0.4592 0.4627
Table 5.15