NO TES
EXAMPLE
Poisson Distribution
In an urban county, health care officials anticipate that the number of births this year will be the same as last year, when 438 children were born—an average of 438y365, or 1.2 births per day. Daily births have been distributed according to the Poisson distribution.
SOLUTION
What Is the Mean of the Distribution?
E(x) 5 can be expressed in a number of ways, because it reflects here the number of occurrences over a span of time. Accordingly, the “span of time”
can be months ( 5 438y12 5 36.5 births per month), weeks ( 5 438y52 5 8.42 births per week), or days ( 5 438y365 5 1.2 births per day). For pur- poses of our example, we will be using 5 1.2 births per day in describing the distribution.
For Any Given Day, What Is the Probability That No Children Will Be Born?
Using the Poisson formula, with 5 1.2 births per day, to find P(x 5 0):
P(x) 5 x ? e2l }x!
and P(0) 5 1.20? e21.2
}0! 5 1 ? 0.30119
}}1 5 0.3012
Calculate Each of the Following Probabilities: P(x 5 1), P(x 5 2), P(x 5 3), P(x 5 4), and P(x 5 5)
Using the same approach as in the preceding calculations, with 5 1.2 mean births per day and values of x from 1 to 5 births per day:
P(x 5 1 birth) 5 1.21• e21.2
}}1! 5 (1.2000)(0.30119)
}}1 5 0.3614
P(x 5 2 births) 5 1.22• e21.2
}}2! 5 (1.4400)(0.30119)
}} 2 • 1 5 0.2169
P(x 5 3 births) 5 1.23• e21.2
}}3! 5 (1.7280)(0.30119)
}} 3 • 2 • 1 5 0.0867
P(x 5 4 births) 5 1.24• e21.2
}}4! 5 (2.0736)(0.30119)
}} 4 • 3 • 2 • 1 5 0.0260
P(x 5 5 births) 5 1.25• e21.2
}}5! 5 (2.4883)(0.30119)
}} 5 • 4 • 3 • 2 • 1 5 0.0062
If we were to continue, we’d find that P(x 5 6) 5 0.0012 and P(x 5 7) 5 0.0002, with both rounded to four decimal places. Including these probabilities
EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EX A
with those just calculated, the result would be the following Poisson probability distribution. As Figure 6.4 shows, the distribution is positively skewed.
x 0 1 2 3 4 5 6 7
P(x) 0.3012 0.3614 0.2169 0.0867 0.0260 0.0062 0.0012 0.0002
What Is the Probability That No More Than One Birth Will Occur on a Given Day?
Since the events are mutually exclusive, this can be calculated as P(x 5 0) 1 P(x 5 1), or 0.3012 1 0.3614, and P(x # 1) 5 0.6626.
Using the Poisson Tables
The Poisson calculations are much easier than those for the binomial. For example, you need compute e2l only one time for a given value of l, since the same term is used in finding the probability for each value of x. Using either the individual or the cumulative Poisson distribution table in Appendix A (Tables A.3 and A.4, respectively) makes the job even easier. Table 6.3 shows a portion of Table A.3, while Table 6.4 shows its cumulative counterpart. If the problem on which you are working happens to include one of the values listed, you need only refer to the P(x 5 k) or the P(x # k) values in the appropriate table.
In the previous example, was 1.2 births per day. Referring to the 5 1.2 column of Table 6.3, we see the same probabilities just calculated for P(x 5 1) through P(x 5 7). For example, in the fifth row of the 5 1.2 column, the entry for P(x 5 4) is 0.0260. To quickly find the probability that there would be no more than 2 births on a given day, we can refer to the cumulative Poisson prob- abilities in Table 6.4 and see that P(x # 2) is 0.8795.
In some cases, the sum of the individual probabilities in a particular Poisson table will not add up to 1.0000. This is due to (1) rounding and (2) the fact that there is no upper limit to the possible values of x. Some values not listed in the table have P(x) .0, but would appear as 0.0000 because of their extremely small chance of occurrence.
0 1 2
x = Number of births during a given day
3 4 5 6 7
0.00 0.10 0.20 0.30 0.40
P(x)
FIGURE 6.4
For the example discussed in the text, the Poisson probability distribution for the number of births per day is skewed to the right. The mean of the distribution is 1.2 births per day, and this descriptor is all that is needed to determine the Poisson probability for each value of x.
PLE EXAMPLE EXAMPLE
Poisson Distribution, Individual Probabilities for x 5 number of occurrences, prob (x 5 k)
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 0 0.3329 0.3012 0.2725 0.2466 0.2231 0.2019 0.1827 0.1653 0.1496 0.1353 1 0.3662 0.3614 0.3543 0.3452 0.3347 0.3230 0.3106 0.2975 0.2842 0.2707 2 0.2014 0.2169 0.2303 0.2417 0.2510 0.2584 0.2640 0.2678 0.2700 0.2707 3 0.0738 0.0867 0.0998 0.1128 0.1255 0.1378 0.1496 0.1607 0.1710 0.1804 4 0.0203 0.0260 0.0324 0.0395 0.0471 0.0551 0.0636 0.0723 0.0812 0.0902 k 5 0.0045 0.0062 0.0084 0.0111 0.0141 0.0176 0.0216 0.0260 0.0309 0.0361 6 0.0008 0.0012 0.0018 0.0026 0.0035 0.0047 0.0061 0.0078 0.0098 0.0120 7 0.0001 0.0002 0.0003 0.0005 0.0008 0.0011 0.0015 0.0020 0.0027 0.0034 8 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002 0.0003 0.0005 0.0006 0.0009 9 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002
10 0.0000 0.0000 0.0000 0.0000
TABLE 6.3 A portion of the table of individual Poisson probabilities from Appendix A.
For 5 1.2 births per day, the probability that there will be exactly 3 births on a given day is P(x 5 3), or 0.0867.
Poisson Distribution, Cumulative Probabilities for x 5 number of occurrences, prob (x k)
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 0 0.3329 0.3012 0.2725 0.2466 0.2231 0.2019 0.1827 0.1653 0.1496 0.1353 1 0.6990 0.6626 0.6268 0.5918 0.5578 0.5249 0.4932 0.4628 0.4337 0.4060 2 0.9004 0.8795 0.8571 0.8335 0.8088 0.7834 0.7572 0.7306 0.7037 0.6767 3 0.9743 0.9662 0.9569 0.9463 0.9344 0.9212 0.9068 0.8913 0.8747 0.8571 4 0.9946 0.9923 0.9893 0.9857 0.9814 0.9763 0.9704 0.9636 0.9559 0.9473 k 5 0.9990 0.9985 0.9978 0.9968 0.9955 0.9940 0.9920 0.9896 0.9868 0.9834 6 0.9999 0.9997 0.9996 0.9994 0.9991 0.9987 0.9981 0.9974 0.9966 0.9955 7 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 0.9996 0.9994 0.9992 0.9989 8 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9998
9 1.0000 1.0000 1.0000 1.0000
TABLE 6.4 A portion of the table of cumulative Poisson probabilities from Appendix A.
For 5 1.2 births per day, the probability that there will be no more than 3 births on a given day is P(x # 3), or 0.9662.
As with the binomial distribution, Excel and Minitab can provide either individual or cumulative Poisson probabilities. The procedures and the results are shown in Computer Solutions 6.3.
The Poisson Approximation to the Binomial Distribution
When n is relatively large and (the probability of a success in a given trial) is small, the binomial distribution can be closely approximated by the Poisson distribution. As a rule of thumb, the binomial distribution can be satisfactorily approximated by the Poisson whenever n $ 20 and # 0.05. Under these con- ditions, we can just use 5n and find the probability of each value of x using the Poisson distribution.
Poisson Probabilities
These procedures show how to get probabilities associated with a Poisson distribution with mean 5 1.2.
For each software package:
Procedure I: Individual or cumulative probabilities for x 5 2 occurrences.
Procedure II: Complete set of individual or cumulative probabilities for x 5 0 occurrences through x 5 8 occurrences.
EXCEL