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
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 m
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
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 () =
l
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 , whi
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(R
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 centr
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 continu
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
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 n