PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
and
(R 1 )
n∑ (X i − µ) 2
σ 2
i=1
∼ χ 2 n
⇒ M 1 (t) =
∴ M 2 (t) =
1
(1 − 2t) n/2 .
1
(1 − 2t) (n−1)/2 ,
which is the mgf for χ 2 n−1.
Hence,
(n − 1)S 2
σ 2
∼ χ 2 n−1.
Corollary.
If X 1 , . . . , X n are i.i.d. N(µ, σ 2 ), then:
Proof.
Recall that t k is the distribution of
Z
√
V/k
, where Z ∼ N(0, 1) and V ∼ χ 2 k independently.
T = ¯X − µ
S/ √ n ∼ t n−1
Now observe that:
and note that,
T = ¯X − µ
S/ √ n = ¯X / √ − µ
σ/ √ (n − 1)S 2 /σ 2
n (n − 1)
¯X − µ
σ/ √ n
∼ N(0, 1)
and
independently.
(n − 1)S 2
σ 2
∼ χ 2 n−1,
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