Final Examination
Statistics 200C
T. Ferguson
June 10, 2010
1. (a) State the Borel-Cantelli Lemma and its converse.
(b) Let X1 , X2 , . . . be i.i.d. from a distribution with density, f (x) = x(+1) on the
a.s.
interval (1, ). For what value of is it true
Final Examination
Statistics 200C
T. Ferguson
June 11, 2009
1. (a) Dene: Xn converges in probability to X .
(b) Dene: Xm converges in quadratic mean to X .
(c) Show that if Xn converges in quadratic mean to X , then Xn converges in probability to X .
2. L
Stat 200C, Spring 2010
Ferguson
Solutions to Exercise Set 10.
24.1. (a) The chi-square is 2 (q1 , q2 , q3 ) =
pij =
q3
q1
q2
r
i=1
c
j =1 (nij
pij )2 /pij , where
if 2 < i < r and 2 < j < c.
if i = 1 and j = 1, or i = 1 and j = c, or i = r and j = 1, or
Stat 200C, Spring 2010
Ferguson
Solutions to Exercise Set 9.
20.5.(a) E (X ) = (1 ) + 2 = 1 + , so the method of moments estimator of is
n = Xn 1.
(b) The log likelihood is
equation is
n ()
=
n () =
log[(1 )exi + (1/2)exi /2 ], so the likelihood
exi /2 +
Stat 200C, Spring 2010
Ferguson
Solutions to Exercise Set 7.
14.1 (a) F (an + bn x)n = (1 + e(an +bnx) )n expcfw_ limn ne(an +bnx) . If we take
bn = 1, we get F (an + bn x)n expcfw_ limn nean ex , which is equal to expcfw_ex =
G3 (x) if we take nean = 1.
Stat 200C, Spring 2010
Ferguson
Solutions to Exercise Set 6.
N
12.4. (a) Write Z EZ = 1 (zj zN )a(Rj ) and T ET similarly. Then, Cov(Z, T ) =
N
N
Cov(Z EZ, T ET ) = 1
1 (zi zN )(tj tN )Cov(a(Ri ), b(Rj ). There are only two
values of Cov(a(Ri ), b(Rj ), n
Stat 200C, Spring 2010
Ferguson
Solutions to Exercise Set 4.
7.8. To nd the asymptotic distribution of 2 = m2 (m1 m3 /m2 ), we need the
asymptotic joint distribution of (m1 , m2 , m3 ). From the central limit theorem with EX =
1 = 0, we have
m1
0
0
L
Stat 200C, Spring 2010
Ferguson
Solutions to Exercise Set 3.
5.5. (a) From the electronic distribution function calculators, we nd P(X 10) =
.58304.
(b) P(X 10) = P(X 10.5) (the correction for continuity). When = 10,
EX = 10 and VarX = 10. So if Z = (X 10
Stat 200C, Spring 2009
Ferguson
Solutions to Exercise Set 1.
1.4. (a) The density of X is fX (x) = x1 I (x > 1). Therefore the expected value of
X is E (X ) = 1 x dx = /( 1). The second moment is E (X 2 ) = 1 x+1 dx =
/( 2). The variance is therefore E (X
Midterm Examination
Statistics 200C
Ferguson
Friday, May 7, 2010
1. Suppose that X has a Poisson distribution with mean , P (), and that Xn is a
L
sequence of random variables such that Xn X . Is it necessarily true that
L
(a) Xn I(Xn > 0) X ?
L
(b) n(Xn