StatisticsPrelim,May 2007
Work all 8 problems.Pleasebegineachproblemon a newsheetof paper,anddo not use
both sidesof the paper(continueon a new sheetof paperif thesolutionwill not fit on a
singlesheet).
from aN(p, o') population,andlet Xzbeanobservation
1. LetXr be an observation
from a N(pt,,to2)population,whereft is a known positiveconstantandXt andXz arc
independent.
(a) Find a completeandsuffrcientstatisticfor (p, o2).
(b) Find the minimumvarianceunbiasedestimatorof p.
(c) Showthat X = (X, + X ,) /2 is an unbiasedestimatorof p. Computethevariance
of .F andthe estimatorfound in part (b). For what valueof ft will the two
variances
be equal?
1,2,.
fr omt h eint eger
s . . ,N .
plitehr e placem ent
2 . L e tX r,..., X n bea ran do m samw
Obtainthemostpowerfultestfor Ho:N: NeversusHr: rV: N1,whereNl > No.
continuous
randomvariables.Provethat
3. LetX and Ibe independent
P(X <Y) = lF,(y)f ,(y)dy, whereFyis thecdf ofXandfyis thepdf of I'.
andidenticallydistributedexponential
4. SupposeXr, X2, andXt areindependent
randomvariableswith meanp > 0. FindP(X3> min(Xr,&)).
5. On observingX from a normaldistributionwith meanp andvariance1,a statistician
interval[X-1.96,X+1.96]. A Bayesianhastheprior belief
declaresthe confidence
p
normal
that is
with mean0 andvariancel/3. Computetheposteriorprobability,as
a functionof X,that p is in the interval.
6. Let the randomvector(X, n havethejoint density
f (x,y I cr,F,T) = ft(o,F,y)*.,-tre-t(l - x - y)'-' ;x > 0,y > 0,.r+ y < 1,u, p,y > 0,
to l.
wherek(a,F, y) is a normalizingconstantsuchthatthe densityintegrates
t
(a) Findthemarginaldistributionof I/. (Hint: Makethe transformation
.)
u= :
l-v'
(b) Findtheconditional
densityofXgiven I'.
i'7
7. SupposeXt, . . ., Xn is a random samplefrom
I
[-rx-utl
,f(xlF,p)=;expl -ff
l;x> p,lr> 0,8> 0.
PLPJ
(a) Findthemethod-of-moments
of prandP.
estimators
(b) Findthemaximum
estimators
of p andP.
likelihood
test
Considerthe
8. Suppose
Xt, ...,Xnisarandom
sample
fromaN(p,o2)population.
ratiotestis equivalent
to the
of H6:p = 0 vs.Hr: F I 0. Showthatthelikelihood
standard
whereS is thesample
usualt-test,whichis to reiectHoif lJ;Xl ) t..r2''-r'
I-fl
deviation, and ro,r r_, is such that P(T,-r > t a/z;n-t)= a l 2 with lr-r a t-random
variablewith n-l degreesof freedom.