264 Some Elementary Statistical Inferences4.4.10.LetY 1 <Y 2 <···<Ynbe the order statistics from a Weibull distribution,
Exercise 3.3.26. Find the distribution function and pdf ofY 1.
4.4.11.Find the probability that the range of a random sample of size 4 from the
uniform distribution having the pdff(x)=1, 0 <x<1, zero elsewhere, is less
than^12.
4.4.12.LetY 1 <Y 2 <Y 3 be the order statistics of a random sample of size 3 from
a distribution having the pdff(x)=2x, 0 <x<1, zero elsewhere. Show that
Z 1 =Y 1 /Y 2 ,Z 2 =Y 2 /Y 3 ,andZ 3 =Y 3 are mutually independent.
4.4.13.Suppose a random sample of size 2 is obtained from a distribution that has
pdff(x)=2(1−x), 0 <x<1, zero elsewhere. Compute the probability that one
sample observation is at least twice as large as the other.4.4.14.LetY 1 <Y 2 <Y 3 denote the order statistics of a random sample of size
3 from a distribution with pdff(x)=1, 0 <x<1, zero elsewhere. LetZ=
(Y 1 +Y 3 )/2 be the midrange of the sample. Find the pdf ofZ.
4.4.15.LetY 1 <Y 2 denote the order statistics of a random sample of size 2 from
N(0,σ^2 ).
(a)Show thatE(Y 1 )=−σ/√
π.
Hint: EvaluateE(Y 1 )byusingthejointpdfofY 1 andY 2 and first integrating
ony 1.(b)Find the covariance ofY 1 andY 2.4.4.16.LetY 1 <Y 2 be the order statistics of a random sample of size 2 from a
distribution of the continuous type which has pdff(x) such thatf(x)>0, provided
thatx≥0, andf(x) = 0 elsewhere. Show that the independence ofZ 1 =Y 1 and
Z 2 =Y 2 −Y 1 characterizes the gamma pdff(x), which has parametersα=1and
β>0. That is, show thatY 1 andY 2 are independent if and only iff(x)isthepdf
of a Γ(1,β) distribution.
Hint: Use the change-of-variable technique to find the joint pdf ofZ 1 andZ 2 from
that ofY 1 andY 2. Accept the fact that the functional equationh(0)h(x+y)≡
h(x)h(y) has the solutionh(x)=c 1 ec^2 x,wherec 1 andc 2 are constants.
4.4.17.LetY 1 <Y 2 <Y 3 <Y 4 be the order statistics of a random sample of size
n= 4 from a distribution with pdff(x)=2x, 0 <x<1, zero elsewhere.
(a)Find the joint pdf ofY 3 andY 4.(b)Find the conditional pdf ofY 3 ,givenY 4 =y 4.(c)EvaluateE(Y 3 |y 4 ).4.4.18. Two numbers are selected at random from the interval (0,1). If these
values are uniformly and independently distributed, by cutting the interval at these
numbers, compute the probability that the three resulting line segments can form
a triangle.