ZNOTES.ORGUPDATED TO 2020-22 SYLLABUSCAIE A2 LEVELMATHSSUMMARIZED NOTES ON THE STATISTICS 2 SYLLABUSCAIE A2 LEVEL MATHSP(A23)=1-P(A<3)1.The Poisson Distribution0The Poisson distribution is used as a model for thenumber,X,of events in a given interval of space or=1-0.690=0.310times.It has the probability formulaPart (ii):P(X=z)=eWrite the distribution using the correct notation(A+B)Po(2(0.65+0.45)=(A+B)Po(2.2)Where A is equal to the mean number of events in the givenintervalUse the limits given in the question to find probabilityA Poisson distribution with mean A can be noted asP(A<4)=e-2.2X~Po(A)=0.8191.2.Suitability of a Poisson Distribution1.5.Relationship of InequalitiesOccur randomly in space or timeOccur singly-events cannot occur simultaneously·P(Xr)=1-P(X≤r)time interval proportional to size of interval1.3.Expectation Variance1.6.Poisson Approximation of aFor a Poisson distribution X~Po (A)Binomial Distribution·Mean=4=E(X)=入·Variance=o2=Var(X)=入To approximate a binomial distribution given by:The mean variance of a Poisson distribution are equalX~B(n,p)1.4.Addition of Poisson Distributions·fn>50 andnp<5Then we can use a Poisson distribution given by:If X and Y are independent Poisson random variables,with parameters A and u respectively,then X+Y has aX~Po(np)Poisson distribution with parameter A+u{IS Ex 8d}Question 8:A randomly chosen doctor in general practice sees,on[IS Ex 8d}Question 1:average,one case of a broken nose per year and each case isThe numbers of emissions per minute from two radioactiveindependent of the other similar cases.objects A and B are independent Poisson variables with1.Regarding a month as a twelfth part of a year,mean 0.65 and 0.45 respectively.Find the probabilities that:1.Show that the probability that,between them,1.In a period of three minutes there are at least threethree such doctors see no cases of a brokenemissions from A.nose in a period of one month is 0.7792.In a period of two minutes there is a total of less than2.Find the variance of the number of cases seenfour emissions from A and B together.by three such doctors in a period of six months2.Find the probability that,between them,three suchSolution:doctors see at least three cases in one year.Part(i):3.Find the probability that,of three such doctors,oneWrite the distribution using the correct notationsees three cases and the other two see no cases inone year.APo(0.65×3)=APo(1.95)Solution:Use the limits given in the question to find probabilityPart(i)(a):Write down the information we know and needWWW.ZNOTES.ORGCAIE A2 LEVEL MATHS1 doctor =1 nose per year =noses per month3 doctors==noses per monthThen we can use a normal distribution given by:Write the distribution using the correct notationX~N(A,A)XPo(0.25)Apply continuity correction to limits:Use the limits given in the question to find probabilityPoissonNormal0.25e-0.255.5≤≤6.5P(X=0)==0.779>6≥6.5≥6≥5.5Use the rules of a Poisson distribution<6≤5.5≤6≤6.5{IS Ex 10h}Question 11:Calculate A in this scenario:The no.of flaws in a length of cloth,lm long has a Poisson入=6×4(in one month)=6×0.25=1.5distribution with mean 0.041.Find the probability that a 10m length of cloth hasfewer than 2 flaws.2.Find an approximate value for the probability that aCalculate in this scenario:1000m length of cloth has at least 46 flaws.3.Given that the cost of rectifying X flaws in a 1000m入=12×4(in one month)=12×0.25=3length of cloth is X2 pence,find the expected cost.Use the limits given in the question to find probabilitySolution:Part(i):P(X≥3)=1-P(X≤2)Form the parameters of Poisson distribution323130l=10andλ=0.04l=1-e-3++0=1-0.423=0.577Write down our distribution using correct notationA for one doctor in one year =1XPo(0.4)入for other two doctors in one year=2×1=2Write the probability required by the questionFor the first doctor:P(X<2)P(X=3)=eFrom earlier equations:For the two other doctors:0.400.410+=0.938P(X=0)=ePart (ii):Using question to form the parametersConsidering that any of the three could be the first1=10andλ=0.04l.入=40>15P(X)=e-ke-13C2=0.025Thus,we can use the normal approximationWrite down our distribution using correct notation1.7.Normal Approximation of a PoissonXPo(40)→YN(40,40)DistributionWrite the probability required by the questionTo approximate a Poisson distribution given by:P(X≥46)X~P(X)Apply continuity correction for the normal distributionWWW.ZNOTES.ORG