Name:
Score:
Math 5080, Statistical Inference I, Fall 2015
Midterm 2
Instructions
1. ‘Write all solutions in the space provided, and use the back pages if you have to.
2. For full credit you must show all work. Providing only an answer will result in very few marks.
3. Calculators are allowed but won’t help much.
4. If you don’t understand the wording of any of the questions feel free to ask.
5. One page of notes is allowed.
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2. (25 points total)
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,
2
Let X
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with A and 8 unknown. Find the method of moments estimators of A and 8. (Hint: how do random variables
from this pdf relate to those with pdf Aex1 {i > O}? Can you use this to easily compute the first and second
moments of such a variable?)
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3. (25 points total)
Let X
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2
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be iid N(, u
) random variables, with
2
i
known but u
2 unknown.
(a) Find the Cramei-Rao lower bound for the variance of an unbiased estimator of u
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2
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2 is unbiased for u
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2
bound? (Hint: recall that the distribution of (n 2
/u is y
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2 1).)
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Extra Credit (5 Points Each)
1. Let N
,.. 5
1
N be independent normals, each with mean zero and E[N?]
E[N] = 9. Write a formula for the number c such that
.
,
E[N]
E[N]
4, E[N]
=
+I\r
(N
+
2
N <c(N+N)) =95
in terms of the percentiles of the F distribution.
r
2. Let X
1
N(O, 12), and X
3
N(O, 4), X
2
do you conWute the probability
N(O, 9), and assume they are all independent of each other. How
(Xi±X2
in terms of the Student’s t-distribution?
1
P
1
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