AMS 553: Homework 3
a < b
, the righttriangular distribution has density function
f
R
(
x
) =
(
2(
x

a
)
(
b

a
)
2
if
a
≤
x
≤
b
0
otherwise.
(1)
and the lefttriangular distribution has density function
f
L
(
x
) =
(
2(
b

x
)
(
b

a
)
2
if
a
≤
x
≤
b
0
otherwise.
(2)
These distributions are denoted by
RT
(
a,b
) and
LT
(
a,b
), respectively.
(a) Show that if
X
∼
RT
(0
,
1), then
X
0
=
a
+ (
b

a
)
X
∼
RT
(
a,b
); verify the same relation between
LT
(0
,
1) and
LT
(
a,b
). Thus it is suﬃcient to generate from
RT
(0
,
1) and
LT
(0
,
1).
(b) Show that if
X
∼
RT
(0
,
1), then 1

X
∼
LT
(0
,
1). Thus it is enough to restrict our attention
further to generating from
RT
(0
,
1).
(c) Derive the inversetransform algorithm for generating from
RT
(0
,
1). Despite the result in (
b
),
also derive the inversetransform algorithm for generating directly from
LT
(0
,
1).
(d) As an alternative to the inversetransform method, show that if
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 Spring '08
 Badr,H
 Probability distribution, Randomness, Cumulative distribution function, Random variate, acceptance rejection method

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