Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.1. Sampling and Statistics 237

(b)Suppose the sampling continues untilX 1 is no longer the smallest observation

(i.e.,Xj<X 1 ≤Xi,i=2, 3 ,...,j−1). LetYequal the number of trials, not

includingX 1 ,untilX 1 is no longer the smallest observation (i.e.,Y=j−1).

Show that the distribution ofYis

P(Y=y)=

1

y(y+1)

,y=1, 2 , 3 ,....

(c)Compute the mean and variance ofYif they exist.

4.1.6.Consider the estimator of the pmf in expression (4.1.10). In equation (4.1.11),
we showed that this estimator is unbiased. Find the variance of the estimator and
its mgf.

4.1.7.The data set on Scottish schoolchildren discussed in Example 4.1.5 included
the eye colors of the children also. The frequencies of their eye colors are

Blue Light Medium Dark

2978 6697 7511 5175

Use these frequencies to obtain a bar chart and an estimate of the associated pmf.

4.1.8.Recall that for the parameterη=g(θ), the mle ofηisg(̂θ), whereθ̂is the
mle ofθ. Assuming that the data in Example 4.1.6 were drawn from a Poisson
distribution with meanλ, obtain the mle ofλand then use it to obtain the mle of
the pmf. Compare the mle of the pmf to the nonparametric estimate. Note: For
the domain value 6, obtain the mle ofP(X≥6).

4.1.9.Consider the nonparametric estimator, (4.1.14), of a pdf.

(a)Obtain its mean and determine the bias of the estimator.

(b)Obtain the variance of the estimator.

4.1.10.This data set was downloaded from the site http://lib.stat.cmu.edu/DASL/
at Carnegie-Melon university. The original source is Willerman et al. (1991). The
data consist of a sample of brain information recorded on 40 college students. The
variables include gender, height, weight, three IQ measurements, and Magnetic
Resonance Imaging (MRI) counts, as a determination of brain size. The data are in
the rda filebraindata.rdaat the sites referenced in the Preface. For this exercise,
consider the MRI counts.

(a)Load the rda filebraindata.rdaand print the MRI data, using the code:

mri <- braindata[,7]; print(mri).

(b)Obtain a histogram of the data,hist(mri,pr=T). Comment on the shape.

(c)Overlay the default density estimator,lines(density(mri)). Comment on

the shape.