APPENDIX B
SOLUTIONS TO PROBLEMS
B.1
Before the student takes the SAT exam, we do not know – nor can we predict with certainty
– what the score will be.
The actual score depends on numerous factors, many of which we
cannot even list, let alone know ahead of time.
(The student’s innate ability, how the student
feels on exam day, and which particular questions were asked, are just a few.)
The eventual SAT
score clearly satisfies the requirements of a random variable.
B.3
(i) Let
Y
it
be the binary variable equal to one if fund
i
outperforms the market in year
t
.
By
assumption, P(
Y
it
= 1) = .5 (a 5050 chance of outperforming the market for each fund in each
year).
Now, for any fund, we are also assuming that performance relative to the market is
independent across years.
But then the probability that fund
i
outperforms the market in all 10
years, P(
Y
i
1
= 1,
Y
i
2
= 1,
…
,
Y
i
,10
= 1), is just the product of the probabilities:
P(
Y
i
1
= 1)
⋅
P(Y
i2
=
1)
…
P(
Y
i
,10
= 1) = (.5)
10
= 1/1024 (which is slightly less than .001).
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 Spring '08
 BREMAN
 Econometrics, Variance, Probability theory

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