1. No category

Homework 3, Sannolikhetsteori
Alex Loiko
910620-1719
Masterprogrammet datalogi, åk. 1
26 oktober 2014
Difference of Exp(1)-distributed values is L(1)-distributed
Show, by using moment generating functions, that if X ∈ L(1),
d
then X = Y1 − Y2 , where Y1 , Y2 ∈ Exp(1) are independent.
If Y1 , Y2 are independent, then Y1 , −Y2 are also independent. The MGF
of Y1 is
Z ∞
ψY1 (t) =
ety fY (y)dy =
0
Z ∞
1 (t−1)y ∞
(t−1)y
e
dy =
e
=
t−1
0
0
1
1
0−
=
t−1
1−t
1
for t < 1. The MGF of −Y2 is then 1+t
. By Theorem 3.2, MGF of sum
1
of random variables, the MGF of Y1 + (−Y2 ) is 1−t
2.
The MGF of a Laplace-distributed s.v. Z ∈ L(1) is
Z ∞
ty 1 −|y|
e
e
dy =
2
−∞
Z
1 ∞ ty−|y|
e
dy =
2 −∞
!
1
1 0
1 ∞
+
=
2
t + 1 −∞
t−1 0
1
1
1
1
+
=
2 t+1 1−t
1 − t2
1
By Theorem 3.1, equal MGFs imply equal distributions Y1 − Y1 ∈
L(1) and the proof is done.
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