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

958 Chapter 29 Statistics and probability distributions
The expected value and variance of the Poisson distributionareboth equal to λ.
Engineeringapplication29.4
Emergencycallstoaserviceengineer
Recordsshowthatonaveragethreeemergencycallsperdayarereceivedbyaservice
engineer. What isthe probability thaton a particular day
(a) three (b) two (c) four
calls will be received?
Solution
Thenumber ofcallsreceived follows aPoissondistribution.Theaverage number of
calls isthreeper day, thatis λ = 3.
(a) P(X =3) = e−3 3 3
= 0.224 (b) P(X = 2) = e−3 3 2
= 0.224
3!
2!
(c) P(X =4) = e−3 3 4
= 0.168
4!
The engineer will receive three calls on approximately 22 days in 100, two calls on
approximately 22 days in100 and four calls on approximately 17 days in100.
Engineeringapplication29.5
Machinebreakdownsinaworkshop
A workshop has several machines. During a typical month two machines will break
down. What arethe probabilitiesthatinamonth
(a) none (b) one (c) morethantwo
will break down?
Solution
λ = average number ofmachinesthat break down = 2
X = numberofmachines brokendown
(a) P(X =0) = e−2 2 0
= 0.135
0!
(b) P(X =1) = e−2 2 1
= 0.271
1!
(c) P(X >2)=1−P(X =0)−P(X =1)−P(X =2)
=1−e −2 −2e −2 −2e −2 =0.323