Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.4. Order Statistics 253

4.3.3.In Exercise 4.2.20, the large sample confidence interval was obtained for the

probability that a tack tossed on a table lands point up. Find the discrete exact

confidence interval for this proportion.

4.3.4.SupposeX 1 ,X 2 ,...,X 10 is a random sample on a random variableXthat
has a Poisson distribution with meanθ. Suppose the realized value of the sample
mean is 0.5; i.e.,nx= 5 events occurred. Suppose we want to compute the exact
90% confidence interval forθ, as determined by equations (4.3.7).

(a)Show that 0.19 and 0.20 bracket the solution to the first equation.

(b)Show that 1.0 and 1.1 bracket the solution to the second equation.

(c)Then use the R functionpoissonci.rto solve the equations.

4.3.5. Consider the same setup as in Example 4.3.1 except now assume thatσ^2
is unknown. Using the distribution of (X−θ)/(S/

√
n), whereSis the sample
standard deviation, set up the equations and derive thet-interval, (4.2.3), forθ.

4.3.6.Using Exercise 3.3.22, show that

∫p

0

n!

(k−1)!(n−k)!

zk−^1 (1−z)n−kdz=

∑n

w=k

(

n

w

)

pw(1−p)n−w,

where 0<p<1, andkandnare positive integers such thatk≤n.
Hint:Differentiate both sides with respect top. The derivative of the right side is
a sum of differences. Show it simplifies to the derivative of the left side. Hence, the
sides differ by a constant. Finally, show that the constant is 0.

4.3.7.This exercise obtains a useful identity for the cdf of a Poisson cdf.

(a)Use Exercise 3.3.5 to show that this identity is true:

λn

Γ(n)

∫∞

1

xn−^1 e−xλdx=

n∑− 1

j=0

e−λ

λj

j!

,

forλ>0andna positive integer.

Hint:Just consider a Poisson process on the unit interval with meanλ.Let

Wnbe the waiting time until thenth event. Then the left side isP(Wn>1).

Why?

(b)Obtain the identity used in Example 4.3.3, by making the transformation

z=λxin the above integral.

4.4 OrderStatistics..............................

In this section the notion of anorder statisticis defined and some of its simple
properties are investigated. These statistics have in recent times come to play an