Page 247 - Elements of Distribution Theory

P1: JZX
052184472Xc07 CUNY148/Severini May 24, 2005 3:59

7.8 Exercises 233

7.21 Let X 1 ,..., X n denote independent random variables such that X j has an absolutely continuous
distribution with density function
−1 α j −1
p j (x j ) = (α j ) x j exp{−x j }, x j > 0
n
where α j > 0, j = 1,..., n. Find the density function of Y = j=1 X j .
7.22 Let X 1 ,..., X n denote independent, identically distributed random variables, each with a stan-
dard exponential distribution. Find the density function of R = X (n) − X (1) .
7.23 Prove Theorem 7.6.
7.24 Let X 1 , X 2 ,..., X n be independent, identically distributed random variables, each with an
absolutely continuous distribution with density function
1
, x > 1.
x 2
Let X ( j) denote the jth order statistic of the sample. Find E[X ( j) ]. Assume that n ≥ 2.
7.25 Let X 1 , X 2 , X 3 denote independent, identically distributed random variables, each with an expo-
nential distribution with mean λ. Find an expression for the density function of X (3) /X (1) .
7.26 Let X 1 ,..., X n denote independent, identically distributed random variables, each with a uni-
form distribution on (0, 1) and let X (1) ,..., X (n) denote the order statistics. Find the correlation
of X (i) and X ( j) , i < j.
7.27 Let X 1 ,..., X n denote independent, identically distributed random variables, each with a stan-
dard exponential distribution. Find the distribution of

n
(X j − X (1) ).
j=1
7.28 Let X = (X 1 ,..., X n ) where X 1 ,..., X n are independent, identically distributed random vari-
ables, each with an absolutely continuous distribution with range X. Let X (·) = (X (1) ,..., X (n) )
denote the vector of order statistics and R = (R 1 ,..., R n ) denote the vector of ranks corre-
sponding to (X 1 ,..., X n ).
n
(a) Let h denote a real-valued function on X . Show that if h is permutation invariant, then
h(X) = h(X (·) ) with probability 1
and, hence, that h(X) and R are independent.
(b) Does the converse hold? That is, suppose that h(X) and R are independent. Does it follow
that h is permutation invariant?
7.29 LetU 1 , U 2 denote independent random variables, each with a uniform distribution on the interval
(0, 1). Let
√
X 1 = (−2 log U 1 ) cos(2πU 2 )
and
√
X 2 = (−2 log U 1 ) sin(2πU 2 ).
Find the density function of (X 1 , X 2 ).
7.30 Consider an absolutely continuous distribution with nonconstant, continuous density function p
and distribution function F such that F(1) = 1 and F(0) = 0. Let (X 1 , Y 1 ), (X 2 , Y 2 ),... denote
independent pairs of independent random variables such that each X j is uniformly distributed
on (0, 1) and each Y j is uniformly distributed on (0, c), where
c = sup p(t).
0≤t≤1