ST 530 February 28, 2005
Name: Midterm #1
All problem parts have equal weight. In budgeting your time expect that some problems will take longer than others.
1. Let X1 , . . . , Xn be an i.i.d. sample from Exponential(1) distribution.
2 2 (a) Fin
Homework #12 solutions
Stat 530, Spring 2010
1. #8.15 We can use the Neymann-Pearson lemma because H0 and H1 are simple. Recall
from a previous problem (and the second exam) that T = n=1 Xi2 is sucient for 2 ,
i
and we also know that T / 2 2 (n), because
Homework #11 solutions
Stat 530, Spring 2010
1. #8.2 Suppose the true rate this year is 15. Then the probability of seeing 10 or fewer
accidents is
10
15k
e15
= .1185,
k!
k=0
which means that it is not very surprising to see as few as 10 accidents. It is
Homework #10 solutions
Stat 530, Spring 2010
1. #7.37 We determined earlier that a sucient statistic for is max(|Xi |). Using
that the cdf for the density for the Xi is FX (x) = ( + x)/(2), we can get the
density for T = max(|Xi |) is
fT (t) =
ntn1
I cfw_
Homework #9 solutions
Stat 530, Spring 2010
1. #4.26 (a)
P (Z z, W = 0) = P (Y z, Y X )
1 z x/ y/
=
e
e
dxdy
0 y
11
=
1 exp z
+
+
and similarly,
P (Z z, W = 1) = P (Y z, Y X ) =
11
1 exp z
+
+
(b) To show Z and W are independent, we rst nd the marginal f
Homework #8 solutions
Stat 530, Spring 2010
1. #7.2 (a) We have
n
L(, |x) =
i=1
1 1 1 xi /
11
xe
=
i
()
()
n
1
n
n
expcfw_
xi
i=1
xi /
i=1
and
n
(, |x) = n log() n log( ) + ( 1) log
n
xi
i=1
xi /
i=1
taking the derivative with respect to and setting
Homework #7 solutions
Stat 530, Spring 2010
1. To show that a statistic T (X ) is not complete for p, we need to nd a function g (t)
where Ep [g (T (X )] = 0 for all p (in this case, for both p). We have S = X1 +X2 +X3
is Binomial(3,p), so S can take valu
Homework #6 solutions
Stat 530, Spring 2010
1. Show that each of the following families of distributions is an exponential family:
(a) Without loss of generality, let = 1. The density is usually written
1
1
1
1
f (x|) = exp (x )2 = exp [x2 2x + 2 ]
2
2
2
Homework #5 solutions
Stat 530, Spring 2010
1. (a) Let X be uniform on (a, b). For a = 4 and b = 5, the true mean and standard
deviation of Y = 1/X are .2231 and .01439, while the estimated values are .2222
and .01426. Not too bad! However, for a = .1 and
Homework #4: solutions
Stat 530, Spring 2010
1. (a) This is easy if you write it down the right way. Write the probabilities in a big
grid as follows:
P (Z n) = P (Z 1)
n=1
P (Z 2)
P (Z 3)
P (Z 4)
P (Z (1, 2]) + P (Z (2, 3]) + P (Z (3, 4]) + P (Z (4, 5])
Homework #3: solutions
Stat 530, Spring 2010
1. The order statistic X(j ) has cdf
n
P (X(j ) x) =
k=j
n
n
k
nk
k
[FX (x)] [1 FX (x)]
=
k=j
n
k
xk [1 x]nk
for the uniform density, if x (0, 1). The median is X(m) where m = (n + 1)/2.
P X(m) (1/2 a, 1/2 + a)
Homework #2: solutions
Spring 2010
1. #5.6 For fun, lets use our three dierent methods. (Each can be done in any
of the three ways we talked about in class.) For (a) Z = X Y , lets use the
conditioning method.
FZ (z ) = P (Z z ) = P (X Y z ) =
=
P (X y +
Homework #13 solutions
Stat 530, Spring 2010
1. #8.55 For part (b), the risk associated with a Type II Error is larger than for part (a).
Note that the risk for > 0 is the same for both parts, but the scales on the plots are
dierent. For = .5, the power i
Homework #14 solutions
Stat 530, Spring 2010
1. #10.1 Lets try the Method of Moments estimator, as its the easiest to nd:
E (X ) =
1
2
1
x(1 + x)dx = ,
3
1
so = 3X . Also we can show that E (X 2 ) = 1/3 for all , so var(X ) = (3 2 )/9.
Show that this is c
ST 530 April 11, 2004
Name: Midterm #2
All problem parts have equal weight. In budgeting your time expect that some problems will take longer than others. You only need to answer 10 out of the 11 problem parts to get full credit. Justify all your
ST 530 April 11, 2004
Name: Midterm #2
All problem parts have equal weight. In budgeting your time expect that some problems will take longer than others. You only need to answer 10 out of the 11 problem parts to get full credit. Justify all your
ST 530 February 28, 2005
Name: Midterm #1
All problem parts have equal weight. In budgeting your time expect that some problems will take longer than others.
1. Let X1 , . . . , Xn be an i.i.d. sample from Exponential(1) distribution.
2 2 (a) Fin
ST 530 May 12, 2005
Name: Final Exam
All problem parts have equal weight. In budgeting your time expect that some problems will take longer than others. Remember, answers without proper justification will not receive full credit!
1. Let X1 , . .
Homework #1 solutions
Stat 530 Spring 2010
1. (a) The random variable Z can take values between 1/b and 1/a. Using the CDF
method, we obtain
FZ (z ) = P (Z z ) = P (U > 1/z )
b 1/z
=
ba
for z (1/b, 1/a). Then
1
.
a)
z 2 (b
5
fZ (z ) =
3
2
0
1
c(0, 5)
4
a