Introduction to Probability and Statistics for Engineers and Scientists

8.7Tests Concerning the Mean of a Poisson Distribution 331

REMARK

If Program 5.2 is not available, one can use the fact that a Poisson random variable
with meanλis, for largeλapproximately normally distributed with a mean and variance
equal toλ.

8.7.1 Testing the Relationship Between Two Poisson Parameters.............

LetX 1 andX 2 be independent Poisson random variables with respective meansλ 1 andλ 2 ,
and consider a test of

H 0 :λ 2 =cλ 1 versus H 1 :λ 2 =cλ 1

for a given constantc. Our test of this is a conditional test (similar in spirit to the Fisher-
Irwin test of Section 8.6.1), which is based on the fact that the conditional distribution
ofX 1 given the sum ofX 1 andX 2 is binomial. More specifically, we have the following
proposition.

PROPOSITION 8.7.1

P{X 1 =k|X 1 +X 2 =n}=

(

n

k

)

[λ 1 /(λ 1 +λ 2 )]k[λ 2 /(λ 1 +λ 2 )]n−k

Proof

P{X 1 =k|X 1 +X 2 =n}

=

P{X 1 =k,X 1 +X 2 =n}

P{X 1 +X 2 =n}

=

P{X 1 =k,X 2 =n−k}

P{X 1 +X 2 =n}

=

P{X 1 =k}P{X 2 =n−k}

P{X 1 +X 2 =n}

by independence

=

exp{−λ 1 }λk 1 /k!exp{−λ 2 }λn 2 −k/(n−k)!

exp{−(λ 1 +λ 2 )}(λ 1 +λ 2 )n/n!

=

n!

(n−k)!k!

[λ 1 /(λ 1 +λ 2 )]k[λ 2 /(λ 1 +λ 2 )]n−k 

It follows from Proposition 8.7.1 that, ifH 0 is true, then the conditional distribution of
X 1 given thatX 1 +X 2 =nis the binomial distribution with parametersnandp=1/(1+c).
From this we can conclude that ifX 1 +X 2 =n, thenH 0 should be rejected if the observed
value ofX 1 , call itx 1 , is such that either

P{Bin(n, 1/(1+c))≥x 1 }≤α/2