Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

236 Some Elementary Statistical Inferences

the histogram. Use the R functiondgamma(x,shape=1,scale=θˆ)to evaluate

the pdf.

(c)Obtain the sample median of the data, which is an estimate of the median

lifetime of a motor. What parameter is it estimating (i.e., determine the

median ofX)?

(d)Based on the mle, what is another estimate of the median ofX?

4.1.2. Here are the weights of 26 professional baseball pitchers; [see page 76 of
Hettmansperger and McKean (2011) for the complete data set]. The data are in R
filebb.rda. Suppose we assume that the weight of a professional baseball pitcher
is normally distributed with meanμand varianceσ^2.

160 175 180 185 185 185 190 190 195 195 195 200 200

200 200 205 205 210 210 218 219 220 222 225 225 232

(a)Obtain a histogram of the data. Based on this plot, is a normal probability

model credible?

(b)Obtain the maximum likelihood estimates ofμ,σ^2 ,σ,andμ/σ. Locate your

estimate ofμon your plot in part (a). Then overlay the normal pdf with these

estimates on your histogram in Part (a).

(c)Using the binomial model, obtain the maximum likelihood estimate of the

proportionpof professional baseball pitchers who weigh over 215 pounds.

(d)Determine the mle ofpassuming that the weight of a professional baseball

player follows the normal probability modelN(μ, σ^2 )withμandσunknown.

4.1.3. Suppose the number of customersXthat enter a store between the hours
9:00 a.m. and 10:00 a.m. follows a Poisson distribution with parameterθ. Suppose
a random sample of the number of customers that enter the store between 9:00 a.m.
and 10:00 a.m. for 10 days results in the values

979151013117212

(a)Determine the maximum likelihood estimate ofθ. Show that it is an unbiased

estimator.

(b)Based on these data, obtain the realization of your estimator in part (a).

Explain the meaning of this estimate in terms of the number of customers.

4.1.4.For Example 4.1.3, verify equations (4.1.4)–(4.1.8).

4.1.5.LetX 1 ,X 2 ,...,Xnbe a random sample from a continuous-type distribution.

(a)FindP(X 1 ≤X 2 ),P(X 1 ≤X 2 ,X 1 ≤X 3 ),...,P(X 1 ≤Xi,i=2, 3 ,...,n).