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Problem Set 4, MGTECON 603, 2009
1
MGTECON 603, Econometrics I
Fall 2009
Stanford GSB
PROBLEM SET 4
Due: Monday October 26
rd
in the CA session
1. Let
X
1
, X
2
, . . ., X
N
be a set of
N <
∞
independent random variables with Gumbel
distributions. The cdf of
X
i
is given by
F
(
x
) = exp (

exp (

x

γ
+
μ
i
))
where
γ
= 0
.
5775 is known as Euler’s constant and
μ
i
is a parameter that is diFerent
for each
i
. The mean of this distribution is given by
μ
i
.
(a) What is the distribution of
Y
1
= max
{
X
1
, ..., X
N
}
?
(b) What is
Pr
(
X
1
≥
max
{
X
2
, ..., X
N
}
)?
2. Let
X
1
, X
2
, . . .
be a sequence of independent random variables with a unit exponential
distribution. Use the delta method to ±nd an approximate normal distribution for
1
/
X
n
.
3. ²ind an example where the sequence
X
1
, X
2
, . . .
converges almost surely but not in
quadratic mean. (You are allowed to use any book you like, but explain it well!!!)
4. Show that if the sequence
X
1
, X
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