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

29.11 The Poisson distribution 957
Isthe new designmore orlessreliable than the
original?
3 Theprobability a machine has a lifespanofmore than
5 years is0.8. Ten machines are chosen atrandom.
What isthe probability that
(a) eightmachines have a lifespanofmore than 5
years
(b) allmachines have alifespan ofmore than 5 years
(c) atleast eight machines have alifespanofmore
than5years
(d) nomore than two machines have alifespanof
lessthan 5 years?
4 Theprobability a valve remainsreliable formore than
10years is 0.75.Eightvalves are sampled. What is
the most likely number ofvalvesto remain reliable
formore than 10years?
5 The probability a chipis manufactured to an
acceptable standard is 0.87.Asampleofsixchipsis
picked at random from a large batch.
(a) Calculate the probability all sixchipsare
acceptable.
(b) Calculate the probability none ofthe chipsis
acceptable.
(c) Calculate the probability that fewer than five
chips in the sampleare acceptable.
(d) Calculate the mostlikely number ofacceptable
chips in the sample.
(e) Calculate the probability that more than two
chips are unacceptable.
Solutions
1 (a) 0.0256 (b) 0.1536
2 (a) 0.9321 (b) 0.0659 (c) 0.9224.
New designis lessreliable
3 (a) 0.3020 (b) 0.1074 (c) 0.678 (d) 0.678
4 6
5 (a) 0.4336 (b) 4.826 ×10 −6 (c) 0.1776 (d) 6
(e) 0.0324
29.11 THEPOISSONDISTRIBUTION
ThePoissondistributionmodelsthenumberofoccurrencesofaneventinagiveninterval.
Consider the number of emergency calls received by a service engineer in one day.
Wemayknowfromexperiencethatthenumberofcallsisusuallythreeorfourperday,
butoccasionallyitwillbeonlyoneortwo,orevennone,andonsomedaysitmaybesix
or seven, or even more. This example suggests a need for assigning a probability to the
numberofoccurrencesofaneventduringagiventimeperiod.ThePoissondistribution
serves this purpose.
Thenumberofoccurrencesofanevent,E,inagiventimeperiodisadiscreterandom
variable which we denote by X. We wish to find the probability that X = 0, X = 1,
X = 2, X = 3, and so on. Suppose the occurrence of E in any time interval is not
affectedbyitsoccurrenceinanyprecedingtimeinterval.Forexample,acarisnotmore,
or less, likely to pass a given spot in the next 10 seconds because a car passed (or did
notpass) the spot inthe previous 10 seconds, thatisthe occurrences areindependent.
Let λ be the expected (mean) value ofX, the number of occurrences during the time
period. IfX is measured for many time periods the average value ofX will be λ. Under
thegivenconditionsX followsaPoissondistribution.TheprobabilitythatX hasavalue
r isgiven by
P(X=r)= e−λ λ r
r!
r=0,1,2,...