Eric Slud
Fall 2007
Solutions to Stat 710 Problem Set 2
#19.3. Z(t) is a standard Brownian motion, which implies that for 0 t 1,
Z(t) tZ(1) Y (t) is a process Gaussian nite dimensional distributions w
Stat 710 HW2 Solutions
# 1 in-class. We are considering iid N (, 2 ) r.v.s Xi , and =
(, ), with generalized method of moments estimators dened in terms of
e() = E (I[X1 1] , I[X1 1] ) = (1 + )/), (1
Eric Slud
Fall 2007
Solutions to Stat 710 Problem Set 3
#6.1. Here we make explicit use of the likelihood ratio exp(Wn ) (dQn /dPn )
= exp(2 /2 + X n ), where X N (0, 1) under Pn . We know tightness o
Stat 710
11/13/02
Solutions to Problem Set 3
Ch. 5 #15. (Xi , Yi ) is iid with joint distribution given by Yi = f0 (Xi )+ei .
Least squares says argmin n1 n (Yi f (Xi )2 . What we are
i=1
minimizing i
Stat 710
12/19/02
Solutions to Selected Problems, HW 5
Ch. 11, #4. The key point in this problem is to use only the projection
denition, with suitable limiting operations, to obtain the usual denition
Eric Slud
Fall 2007
Solutions to Stat 710 Problem Set 4
#7.1. Here the dominating measure on N = cfw_0, 1, 2, . . . is counting measure.
For each x 0, the function
s(x, ) p(x, ) = exp(x log )/2)/ x!
i
1.
A2 ~ N ( 30, 9 )
A1 ~ N ( 0, 12 )
9
12
B ~ N ( 70, 20 )
20
T = Time (mins)
elapsed from 1:00
1:00
2:00
3:00
(a) Determine the distribution of the variable A = Amount of time for Al to reach the con
Solutions for HW1, Stat 710, F07
#5.12. The condition is: there exists a unique pth quantile xp , i.e. a number
such that P (X1 xp ) = p and for all > 0,
P (X1 xp ) < p,
P (X1 xp + ) < 1 p
The Lemma n
1. (a) If = .05, then /2 = .025, so that z.025 = 1.96. If 1 = .999, then = .001, so that z.001 = 3.09.
2
| 3000 2500 |
1.96 + 3.09
Moreover,
= = 0.625 , so that n =
= 65.29, so take n 66.
800
0.
1. Given: Independent temperature distributions (F) of two islands X1 ~ N(80, 3) and X2 ~ N(83, 4).
(a) P(80 X1 83) = P(X1 83) P(X1 80)
= P(Z 1) P(Z 0)
since 83 here corresponds to a z-score of 83 80
Stat 710 HW1 Solutions
(2). The density of the tn distributed random variable Xn = Zn / Vn
has density of the form Cn (1 + x2 /n)(n+1)/2 , where Zn N (0, 1) and
Vn Gamma(n/2, n/2) are independent. We