PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
If X ∼Exp(λ) and Y = log X, then we can find f Y (y) by taking h(x) = log x and using
f Y (y) = f X (h −1 (y))|h −1 (y) ′ | [h −1 (y) = e y , h −1 (y) ′ = e y ]
= λe −λey e y
{ n
}
∏
=⇒ l Y (λ; y) = log λe −λey i
e y i
= log
i=1
{
λ n e −λ P n
i=1 ey i
e P }
n
i=1 y i
= n log λ − λ
=⇒ ∂
∂λ l Y (λ; y) = n n∑
λ −
=⇒ ˆλ =
=
=
= 1¯x
n
n∑
i=1
i=1
e y i
n
n∑
i=1
n
n∑
i=1
e log x i
x i
e y i
n∑
e y i
+
i=1
( n
n¯x)
.
n∑
i=1
y i
Finally, suppose we take θ = log λ =⇒ λ = e θ
=⇒ f(x; θ) = e θ e −eθ x
(λe −λx )
( n
)
∏
=⇒ l θ (θ; x) = log e θ e −eθ x i
= log
i=1
{ }
e nθ e −eθ n¯x
= nθ − e θ n¯x.
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