Practice Midterm Exam I
Math 362 Name:
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Math 362, Problem set 1
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,
We have that:
E[X] = E[
nE[Xi ] = E[Xi ]
Math 362, Problem set 3
1. (5.4.18) Using the assumptions behind the condence interval given in
expression (5.4.17), show that
+ 2 P 1.
The real key here is to address why the assumption n1 /n 1 and
Math 362, Problem set 5
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
1. (6.1.2) Let X1 , X2 , . . . , Xn be a random sample from a ( = 3, = )
distribution, 0 < < . Determine the mle of .
() = 3n log() n log(2) +
2 log(Xi )
Math 362, Problem set 8
Due 4/13/10 (okay to turn in on 4/15)
Answer: We have that the mle = Yn . We have
if Yn .
( Yn )n
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
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 .
fX1 ,.,Xn (x1 , . . . , xn ; ) = en
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 .