PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
where φ(a) = Φ ′ (a) = 1 √
2π
e −a2 /2 is the N(0, 1) PDF
= 1 2 x−1/2 [ 1
√
2π
e −x/2 + 1 √
2π
e −x/2 ]
(
= (1/2)1/2
1
√ x 1/2−1 e −1/2x, which is the Gamma
π 2 , 1 2)
PDF.
On the other hand, the distribution of Z 2 is also called the χ 2 1, distribution. We
have proved that it is the same as the Gamma ( 1
2 , 1 2)
distribution.
3. Moments of transformed RVS:
If U ∼ U(0, 1) and Y = − log U
λ
then Y ∼ Exp(λ) ⇒ f(y) = λe −λy , y > 0.
Can check
E(Y ) =
∫ ∞
0
λye −λy dy
= 1 λ .
4. Based on Theorem 1.6.1:
If U ∼ U(0, 1) and Y = − log U , then according to theorem 1.6.1,
λ
E(Y ) =
∫ 1
0
= − 1 λ
= − 1 λ
− log u
(1) du
λ
∫ 1
0
log u du = − 1 λ (u log u − u) ∣ ∣∣∣
1
[
u log u ∣ ∣1
0 − u ∣ ]
1 0
0
= − 1 [0 − 1]
λ
= 1 , as required.
λ
There are some important consequences of Theorem 1.6.1:
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