30.16 EXERCISES
according to one of the following schemes, which have been approved for the
purpose.
(a) The entire carton is weighed, and the vendor is fined 2500 euros if the
average weight of a kipper is less than 0.1975 kg.
(b) Twenty-five kippers are selected at random from the carton; the vendor is
fined 100 euros if the average weight of a kipper is less than 0.1980 kg.
(c) Kippers are removed one at a time, at random, until one has been found
that weighsmorethan 0.2000 kg; the vendor is fined 4n(n−1) euros, where
nis the number of kippers removed.
Which scheme should the Chancellor of the Exchequer be urging his government
to adopt?
30.32 In a certain parliament, the government consists of 75 New Socialites and
the opposition consists of 25 Preservatives. Preservatives never change their
mind, always voting against government policy without a second thought; New
Socialites vote randomly, but with probabilitypthat they will vote for their party
leader’s policies.
Following a decision by the New Socialites’ leader to drop certain manifesto
commitments,Nof his party decide to vote consistently with the opposition. The
leader’s advisors reluctantly admit thatan election must be called ifNis such
that, at any vote on government policy, the chance of a simple majority in favour
would be less than 80%. Given thatp=0.8, estimate the lowest value ofNthat
would precipitate an election.
30.33 A practical-class demonstrator sends his twelve students to the storeroom to
collect apparatus for an experiment, but forgets to tell each which type of
component to bring. There are three types,A,BandC, held in the stores (in
large numbers) in the proportions 20%, 30% and 50%, respectively, and each
student picks a component at random. In order to set up one experiment, one
unit each ofAandBand two units ofCare needed. Let Pr(N) be the probability
that at leastNexperiments can be set up.
(a) Evaluate Pr(3).
(b) Find an expression for Pr(N)intermsofk 1 andk 2 , the numbers of compo-
nents of typesAandBrespectively selected by the students. Show that Pr(2)
canbewrittenintheform
Pr(2) = (0.5)^12∑^6
i=2(^12) Ci(0.4)i
∑^8 −i
j=2
12 −iCj(0.6)j.
(c) By considering the conditions under which no experiments can be set up,
show that Pr(1) = 0.9145.
30.34 The random variablesXandYtake integer values,xandy, both≥1, and such
that 2x+y≤ 2 a,whereais an integer greater than 1. The joint probability
within this region is given by
Pr(X=x, Y=y)=c(2x+y),
wherecis a constant, and it is zero elsewhere.
Show that the marginal probability Pr(X=x)is
Pr(X=x)=
6(a−x)(2x+2a+1)
a(a−1)(8a+5)
,
and obtain expressions for Pr(Y=y), (a) whenyis even and (b) whenyis odd.
Show further thatE[Y]=6 a^2 +4a+1
8 a+5