082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

Table29.6
Theprobabilitiesforbinomial and Poissondistributions.
29.11 The Poisson distribution 959
Binomial (n,p) Poisson (λ)
P(X =r);n =15,p=0.05 P(X =r); λ =0.75
r = 0 0.46339 0.47237
r = 1 0.36576 0.35427
r = 2 0.13475 0.13285
r = 3 0.03073 0.03321
r = 4 0.00485 0.00623
r = 5 0.00056 0.00090
r = 6 0.00005 0.00007
r = 7 0.00000 0.00001
29.11.1 Poissonapproximationtothebinomial
ThePoissonandbinomialdistributionsarerelated.Consider abinomialdistribution,in
whichntrialstakeplaceandtheprobabilityofsuccessisp.Ifnincreasesandpdecreases
such that np is constant, the resulting binomial distribution can be approximated by a
Poisson distribution with λ = np. Recall thatnp is the expected value of the binomial
distribution,and λisthe expected value of the Poisson distribution.
Toillustratetheabovepoint,Table29.6liststheprobabilitiesforbinomialandPoissondistributionswithn
= 15,p= 0.05andhence λ = 15(0.05) = 0.75.Theremaining
probabilities are all almost 0.
Asnincreases and p decreases withnp remaining constant, agreement between the
two distributions becomes closer.
Engineeringapplication29.6
Workforceabsentees
A workforce comprises 250 people. The probability a person is absent on any one
day is0.02. Find the probability thaton a day
(a) three
people are absent.
(b) seven
Solution
This problem may be treated either as a sequence of Bernoulli trials or as a Poisson
process.
Bernoullitrials
The probabilities followabinomial distribution.
E:apersonis absent
n =number oftrials = 250
p = probability thatE occurs inasingletrial =0.02
X = numberof occurrencesof eventE
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