Section 5.7
PROPERTIES OF A RANDOM SAMPLE
256
(deFinetti proved an elegant characterization theorem for an infinite sequence of ex.
changeable random variables. He proved that any such sequence of exchangeable ran.
dom variables is a mixture of iid random variables.)
5.5 Let Xl , . . . , X" be iid with pdf fx (x), and let
X denote the sample mean. Show that
even if the mgf of X does not exist.
5.6 If X has pdf fx (x) and Y, independent of X, has pdf jy (y), establish formulas, simi lar
to
(5.2.3), for the random variable Z in each of the following situations.
(a) Z
=
X-Y
(b) Z
XY
(c) Z
X/Y
5.2. 10, a partial fraction decomposition is needed to derive the distribution
of the sum of two independent Cauchy random variables. This exercise provides the
details that are skipped in that example.
5.1 In Example
(a) Find the constants A, B, C, and D that satisfy
1
1
1 + (W/ U) 2 1 + « z - w)/r) 2
D
B
Cw
Aw
+
2
1
+
1
+
(w/
«
z
+
1 (w/a)
aF
� w)/r) 2
1 + « z - w)/r)2 '
where A, B, C, and D may depend on z but not on w.
(b) Using the facts that
J 1:
t2
evaluate
dt =
�
log ( 1 + t2 ) + constant and
J
� dt
l + t2
=
arctan(t) + constant,
(5.2.4) and hence verify (5.2.5).
(Note that the integration in part (b) is quite delicate. Since the mean of a Cauchy
dw do not exist.
dw and
does not exist, the integrals
However, the integral of the difference does exist, which is all that is needed.)
J::::' 1+(:/'0")2
J�oo 1+« .°:)/1')2
Section 5.7
&.8 Let
257
EXERCISES
Xl , . . " X" be a ra.ndom sa.mple, where X a.nd 82 are calculated in the usual way.
(a) Show that
82 =
n
1
n
X
1) L 2 ) i
2n(n
i=l j = l
Assume now that the XiS have a finite fourth moment, and denote
E ( X, - (1 ) 1 , j = 2 , 3 , 4
(b) Show that Var 82 = � (04
.
�=�O�).
(c) Find Cov(X, 82) i n terms oUI ,
• • .
,
, 04 • Under what conditions is Cov (X, 82)
5 .9 Establish the Lagrange Identity, that for any numbers aI , a2
, a n and
. . ·
Use the identity t o show that the correlation coefficient is equal t o
of the sample points lie on a straight line (Wright
for
..
n = 2;
then induct.)
1992). (Hint:
1
OJ
=
= a?
bl , b2 , . . . , bn ,
i f a.nd only i f all
Establish the identity
Xl , . , Xn be a ra.ndom sample from a n(J1., (72) population.
(a) Find expressions for 0 1 ,
, 04 , as defined in Exercise 5.8, in terms of J1. and (72.
(b) Use the results of Exercise 5.8, together with the results of part (a) , to calculate
Var 82•
(c) Calculate Var 82 a completely different (and easier) way: Use the fact that
(n 1)82/(72 X�-l '
Suppose X a.nd 8 2 are calculated from a random sample X l ,
, X drawn from a
population with finite variance (J"2 . We know that E82
(J"2. Prove that E8 $ (J", and
if (72 > 0, then E8 < (7.
5.10 Let
• . •
'"
5.11
(it = EX"
.
. . ..
5.12 Let X l , " " Xn be a random sample from a nCO, 1 ) population. Define
Calculate EY1 and EY2 , a.nd establish an inequality between them.
5 . 13 Let
Xl , . . , X". be iid n(J1., (72). Find a function of 82, the sample variance, say g(82),
Eg(82) (J". ( Hint: Try g(82) c..(S2, where c is a constant. )
Prove that the statement o f Lemma 5.3.3 follows from t h e special case o f J1. i = a
and (7[ = 1 . That is, show that if Xj
(7) Zj + J1.j and Zj "'"' n(O, l ) , j = 1, . , n,
all independent; aij , brj are constants, and
.
that satisfies
5.14 (a)
..
then
n
a =}
Laij Xj and
j =l
n
LbrjXj are independent.
j =l
( b) Verify the expression for Cov ( 2:::;= 1 aijXj , 2:::;= 1 brj Xj
)
in Lemma
5.3.3.
PROPERTIES OF A RANDOM SAMPLE
2&8
Section 5.7
5.15 Establish the following recursion relations for means and variances. Let X.,. and 8! be
the mean and variance, respectively, of Xl , . . . , Xn• Then suppose another observation,
Xn + l , becomes available. Show that
_ x n + l + nX',,(a) Xn+1 n +1
_
- 2
.
" 1 ) (Xn+ 1 - Xn)
(b) n8n2 + l - (n - 1 ) 8..2 + ( n +
1 , 2, 3, be independent with n(i , i 2 ) distributions. For each of the following
situations, use the XiS to construct a statistic with the indicated distribution.
5.16 Let Xi, i =
(a) chi squared with 3 degrees of freedom
(b) t distribution with 2 degrees of freedom
(c) F distribution with 1 and 2 degrees of freedom
X be a random variable with an Fp.q distribution.
(a) Derive the pdf of X .
(b) Derive the mean and variance o f X .
(c) Show that I /X has an Fq.p distribution.
(d) Show that (p/q)X/[1 + (p/q)X] has a beta distribution with parameters p/2 and
q/2.
5.18 Let X be a random variable with a Student's t distribution with p degrees of freedom.
5.17 Let
( a) Derive the mean and variance of X .
(b) Show that X 2 has an F distribution with 1 and p degrees of freedom.
(c) Let f(xlp) denote the pdf of X. Show that
at each value of x, -00 < x < 00. This correctly suggests that as p -> 00, X con­
verges in distribution to a nCO, 1 ) random variable. (Hint: Use Stirling's Formula.)
(d) Use the results of parts (a) and (b) to argue that, as p -> 00, X 2 converges in
distribution to a xi random variable.
(e) What might you conjecture about the distributional limit, as p -> 00, of qF",p?
5.19 (a) Prove that the X2 distribution is stochastically increasing in its degrees of freedom;
that is, if p > q, then for any a, P(X� > a ) ;::: P(X; > a ) , with strict inequality for
some a.
(b) Use the results of part (a) to prove that for any II, kFk,u is stochastically increasing
in k.
(c) Show that for any k, II, and a:, k Fa,k,,, > ( k - I) Fa,k- 1 ,,, , (The notation Fa, k - l ,,,
denotes a level-a: cutoff point; see Section 8.3.1. Also see Miscellanea 8.5.1 and
Exercise 1 1 .15,)
5.20 (a) We can see that the t distribution is a mixture of normals using the following
argument:
�P (Tu S t) = P
( � ) roo
v xUv
st
=
10
P (z s ty'i/JiI) P (x� = x) dx,