Practice Midterm Exam I
Math 362 Name:
2/25/10
Read all of the following information before starting the exam:
0 READ EACH OF THE PROBLEMS OF THE EXAM CAREFULLY!
0 Show all work, clearly and in order7 if you want to get full credit. I reserve the right to
1. (25 points) Suppose X1, X2, . . .,Xn is a sample from a E$p0(1/6) distribution, and let
Y1, . . . ,Yn denote the order statistics of the sample. a. (10 pts) Find the constant c so
that 0Y1 is an unbiased estimator of (9; '
restsas Fkaig%g%iymwe
gig {i
1. (20 points)
We take a sample of size 2 from a Poisson distribution with parameter 0; the pmf of this
distribution is p(k; 0) = P0(X = k) = 8—1:“.
We test H0 : 0 = 1 versus H1 : 9 = 2, by accepting H1 if
P(X1; 2)}?(X2; 2)
p(X1;1)P(X2;1)
>2.
a. (10 pts
APPM/MATH 4/5520
Solutions to Exam I Review Problems
1. (a)
fX1 ,X2 (x1 , x2 ) dx2
fX1 (x1 ) =
x1 x2
x1 2e
=
dx2
= 2e2x1
x1 was below x2 , but when marginalizing out x2 , we ran it over all values from 0 to
and so there was no upper bound on x1 . The na
Practice Midterm Exam II
Math 362 Name:
2/25/10
Read all of the following information before starting the exam:
0 READ EACH OF THE PROBLEMS OF THE EXAM CAREFULLY!
0 Show all work, clearly and in order, if you want to get full credit. I reserve the right
APPM/MATH 4/5520
Solutions to Exam II Review Problems
1.
FYn (y) = P (Yn y) = P (n ln(X(1) + 1) y)
= P (X(1) ey/n 1)
Since
P (X(1) x)
=
1 P (X(1) > x)
iid
1 [P (X1 > x)]n
=
P areto
=
1
(1+x)
=
we have that
1
1
n
1
(1+x)n ,
FYn (y) = P (X(1) ey/n 1)
= 1
1
APPM/MATH 4/5520
Exam I Review Problems
Exam I: Thursday, October 2nd from 6:30 to 9pm in FLMG 155.
Optional Extra Review Session: Wednesday, October 1st from 7 to 9 pm in ECCR 110.
The actual exam will have 6 problems and you will only have to do 5 of th
APPM/MATH 4/5520
Exam II Review Problems
Exam I: will be on Thursday, November 6th from 6:30 to 9:00pm in FLMG 155.
Optional Extra Review Session: will be on Wednesday, November 13th from 6 to 8 pm in
ECCR 110.
1. Let X1 , X2 , . . . , Xn be a random samp