TA session 4
Econ. 103, winter 2010
Wed., Jan. 27, 2010, 10:00 a.m. and 1:00 p.m. in PP2400E.
2
The distribution of
X
i
is not the distribution of
X
This is something people have trouble with at ﬁrst. Let
X
1
∼ N
(
μ,σ
2
)
Suppose
μ
= 0 and
σ
= 1. Then the distribution of
X
1
looks something like
Now, in addition to
X
, let
X
2
∼ N
(
μ,σ
2
)
Note that
X
2
has the same mean and variance as
X
1
!
From the rules I gave you in my week 3 lecture, you know that
Var
±
X
1
+
X
2
2
²
=
σ
2
2
2
+
σ
2
2
2
=
σ
2
2
Hence (
X
1
+
X
2
)
/
2
∼ N
(
μ,σ
2
/
2):
1
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Moral of the story
The variance of several random variables with the same distribution added together is
X
1
+
X
2
+
X
3
+
...
+
X
n
n
∼ N
(
μ,σ
2
/n
)
and if one were to graph this distribution, it would tend to collapse around the mean as
n
→ ∞
. Here are several distributions with
μ
= 0 using
n
= 1
,
2
,
4
,
8:
Now the estimate of the mean is just
X
i
=
∑
N
n
=1
X
i
n
and having endured the above, you should recognize its distribution
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 Winter '07
 SandraBlack
 Econometrics, Standard Deviation, Variance, Mean, Probability theory, probability density function

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