Practice Midterm Exam I
Math 362 Name:
2/25/10
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0 READ EACH OF THE PROBLEMS OF THE EXAM CAREFULLY!
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Math 362, Problem set 1
Due 1/31/10
1. (4.1.8) Determine the mean and variance of the mean X of a random
sample of size 9 from a distribution having pdf f (x) = 4x3 , 0 < x < 1,
zero elsewhere.
Answer:
We have that:
E[X] = E[
1
n
1
nE[Xi ] = E[Xi ]
n
Xi
Math 362, Problem set 3
Due 2/16/10
1. (5.4.18) Using the assumptions behind the condence interval given in
expression (5.4.17), show that
2
S1
S2
+ 2/
n1
n2
2
1
2
+ 2 P 1.
n1
n2
Answer:
The real key here is to address why the assumption n1 /n 1 and
2
2
n
Math 362, Problem set 5
Due 3/16/10
1. (3.7.6) Another estimating chi-square: Let the result of a random experiment be classies as one of the mutually exclusive and exhaustive ways
A1 , A2 , A3 and also as one of the mutually exclusive and exhaustive ways
Math 362, Problem Set 6
Due 3/23/11
1. (6.1.2) Let X1 , X2 , . . . , Xn be a random sample from a ( = 3, = )
distribution, 0 < < . Determine the mle of .
Answer
We have
L() =
1
3n 2n
2
(Xi )e
Xi /
() = 3n log() n log(2) +
() =
Solving
3n
+
Xi
2
2 log(Xi )
Math 362, Problem set 8
Due 4/13/10 (okay to turn in on 4/15)
1. (6.3.18)
Answer: We have that the mle = Yn . We have
L() =
1
.
n
if Yn .
Computing
=
( Yn )n
0
0
L(0 )
=
L(Yn )
if Yn 0
if Yn > 0 .
For (b), if Z = 2 log = 2n log(Yn /0 ), then Yn = 0 eZ/2n
Math 362, Problem set 9
Due 4/20/10
1. (7.2.2) Prove that the sum of the observations of a random sample of size
n from a Poisson distribution of having parameter , 0 < < , is a
sucient statistic for .
Answer:
fX1 ,.,Xn (x1 , . . . , xn ; ) = en
and so
x
Math 362, Problem set 10
Due 4/25/10 (okay to turn in on 4/27 - I will be traveling, though)
1. (7.5.10) Let X1 , . . . , Xn be a random sample from a distribution of pdf
f (x; ) = 2 xex .
(a) Argue that Y =
Xi is a complete sucient statistic for .
(b) Co