PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Consider the identity:
n∑
n∑
(X i − µ) 2 = (X i − ¯X) 2 + n( ¯X − µ) 2
i=1
i=1
(subtract and add ¯X to first term)
⇒
n∑
i=1
(X i − µ) 2
σ 2 =
n∑
i=1
(X i − ¯X) 2
σ 2 +
( ) 2 ¯X − µ
σ/ √ n
and let R 1 =
R 2 =
n∑ (X i − µ) 2
i=1
n∑
i=1
σ 2
(X i − ¯X) 2
σ 2 =
(n − 1)S2
σ 2
R 3 =
( ) 2 ¯X − µ
σ/ √ n
If M 1 , M 2 , M 3 are the MGFs for R 1 , R 2 , R 3 respectively, then,
M 1 (t) = M 2 (t)M 3 (t) since R 1 = R 2 + R 3
with R 2 and R 3 independent
↓ ↓
depends depends only
only on on ¯X
Next, observe that R 3 =
( ) 2 ¯X − µ
σ/ √ ∼ χ 2 1
n
S 2
⇒ M 2 (t) = M 1(t)
M 3 (t) .
⇒ M 3 (t) =
1
(1 − 2t) 1/2 ,
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