PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Examples
1. Suppose X is continuous with CDF F (x), and F (a) = 0, F (b) = 1; (a, b can be
±∞ respectively).
Let U = F (X). Observe that
M U (t) =
∫ b
a
e tF (x) f(x) dx
which is the U(0, 1) MGF.
= 1 t etF (x) ∣ ∣∣∣
b
= et − 1
,
t
2. Suppose X ∼ U(0, 1), and let Y = − log X .
λ
M Y (t) =
∫ 1
0
a
e
= etF (b) tF (a)
− e
t
− log x
t( λ
) (1) dx
=
=
=
∫ 1
0
x −t/λ dx
∣
1 ∣∣∣
1
1 − t/λ x1−t/λ
1
1 − t/λ
0
= λ
λ − t ,
which is the MGF for Exp(λ) distribution. Hence we can conclude that
Y = − log X
λ
∼ Exp(λ).
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