IEOR E4703: Practice Midterm Exam, Fall 2010. Professor Sigman.
1.
X
1
and
X
2
are two independent random variables distributed as:
P
(
X
1
= 0) = 0
.
30
, P
(
X
1
= 1) = 0
.
50
, P
(
X
1
= 2) = 0
.
20 and
P
(
X
2
= 1) = 0
.
40
, P
(
X
2
=
3) = 0
.
60
(a) Give an algorithm for generating from
X
=
X
1
X
2
that uses two uniform numbers
U
1
,
U
2
.
(b) Give an algorithm for generating from
X
=
X
1
X
2
that uses only one uniform number
U
.
(c) Suppose that
X
has an exponential distribution at rate
λ
= 2;
F
(
x
) =
P
(
X
≤
x
) =
1

e

2
x
. Suppose you wish to generate a rv
Y
that has the conditional distribution
of
X
given that it falls in the interval (4
,
10).
Show that the following algorithm
works:
i. Generate
V
uniformly distributed over the interval (
c, d
) where
c
=
F
(4) and
d
=
F
(10).
ii. Set
Y
=

(1
/
2) ln(1

V
).
2. Consider a 2dimensional BM,
X
∼
BM
(
μ,
Σ
), where
Σ
=
σ
2
1
σ
1
σ
2
ρ
σ
1
σ
2
ρ
σ
2
2
!
.
Furthermore, independent of
X
, consider a compound Poison process,
Y
(
t
) =
N
(
t
)
X
i
=1
J
i
,
(1)
where
{
N
(
t
)
}
is a Poisson process at rate
λ
, and the jumps
{
J
i
}
are iid distributed as the
standard
double exponential
:
f
(
x
) =
e

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 Spring '07
 sigman
 Probability theory, x1 x2

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