Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.4. Order Statistics 263

4.4.3.Consider the sample of data (data are in the fileex4.4.3data.rda):

13 5 202 15 99 4 67 83 36 11 301

23 213 40 66 106 78 69 166 84 64

(a)Obtain the five-number summary of these data.

(b)Determine if there are any outliers.

(c)Boxplot the data. Comment on the plot.

4.4.4.Consider the data in Exercise 4.4.3. Obtain the normalq−qplot for these

data. Does the plot suggest that the underlying distribution is normal? If not, use

the plot to determine a more appropriate distribution. Confirm your choice with a

q−qbased on the quantiles using your chosen distribution.

4.4.5.LetY 1 <Y 2 <Y 3 <Y 4 be the order statistics of a random sample of size
4 from the distribution having pdff(x)=e−x, 0 <x<∞, zero elsewhere. Find
P(Y 4 ≥3).

4.4.6.LetX 1 ,X 2 ,X 3 be a random sample from a distribution of the continuous
type having pdff(x)=2x, 0 <x<1, zero elsewhere.

(a)Compute the probability that the smallest ofX 1 ,X 2 ,X 3 exceeds the median

of the distribution.

(b)IfY 1 <Y 2 <Y 3 are the order statistics, find the correlation betweenY 2 and

Y 3.

4.4.7.Letf(x)=^16 ,x=1, 2 , 3 , 4 , 5 ,6, zero elsewhere, be the pmf of a distribution

of the discrete type. Show that the pmf of the smallest observation of a random

sample of size 5 from this distribution is

g 1 (y 1 )=

(

7 −y 1

6

) 5

−

(

6 −y 1

6

) 5

,y 1 =1, 2 ,..., 6 ,

zero elsewhere. Note that in this exercise the random sample is from a distribution

of the discrete type. All formulas in the text were derived under the assumption

that the random sample is from a distribution of the continuous type and are not

applicable. Why?

4.4.8.LetY 1 <Y 2 <Y 3 <Y 4 <Y 5 denote the order statistics of a random sample
of size 5 from a distribution having pdff(x)=e−x, 0 <x<∞, zero elsewhere.
Show thatZ 1 =Y 2 andZ 2 =Y 4 −Y 2 are independent.
Hint: First find the joint pdf ofY 2 andY 4.

4.4.9.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample of sizen
from a distribution with pdff(x)=1, 0 <x<1, zero elsewhere. Show that the
kth order statisticYkhas a beta pdf with parametersα=kandβ=n−k+1.