PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
2. Independence of ¯X and S: consider
⎡
⎤
1 1
1 1
. . .
n n n n
1 − 1 − 1 . . . − 1 − 1 n n n n
A =
− 1 1 − 1 . . . − 1 − 1 n n n n
⎢
.
.. .
.. .
.. . .
⎥
⎣
⎦
− 1 − 1 . . . 1 − 1 − 1 n n n n
and observe
⎡ ⎤
¯X
X 1 − ¯X
AX =
X 2 − ¯X
⎢ ⎥
⎣ . ⎦
X n−1 − ¯X
We can check that Var(AX) = σ 2 AA T has the form:
⎡
⎤
1
0 . . . 0
n 0
σ 2 ⎢
. Σ 22
⎥
⎣
⎦
0
⇒ ¯X, (X 1 − ¯X, X 2 − ¯X, . . . , X n−1 − ¯X) are independent.
(By multivariate normality.)
Finally, since
X n − ¯X = − ( (X 1 − ¯X) + (X 2 − ¯X) + · · · + (X n−1 − ¯X) ) ,
it follows that S 2 is a function of (X 1 − ¯X, X 2 − ¯X, . . . , X n−1 − ¯X) and hence is
independent of ¯X.
3. Prove:
(n − 1)S 2
σ 2
∼ χ 2 n−1
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