IEOR 4703: Homework 7
1. Consider the problem of estimating
θ
(
x
) =
E
[
e
√
Z
I
{
Z
≥
x
}
]
for
x
≥
1 and where
Z
∼
N(0
,
1). Let
X
:=
e
√
Z
I
{
Z
≥
x
}
.
(a) Show that
θ
(
x
)
≥
(1

Φ(
x
))
e
√
x
where Φ(
.
) is the CDF of a standard normal
random variable.
(b) Show that
E
[
X
2
]
≥
(1

Φ(
x
))
e
2
√
x
(c) Consider doing importance sampling with alternative density
g
(
x
) having the
density of
W
∼
N(
x,
1). (That is, we shift the mean by
x
.)
Show that
θ
(
x
) =
E
[
Y
] where
Y
=
e
x
2
/
2
e
√
W
e

Wx
I
{
W>x
}
.
Further show that
E
[
Y
]
≤
e

x
2
/
2
E
[
e
√
W
I
{
W>x
}
]
.
(d) Using the fact that
W
may be expressed as
Z
+
x
, show that
θ
(
x
)
≤
e
1

(
x

1)
2
/
2
.
(In particular
θ
(
x
)
→
0 as
x
→ ∞
, which can also be proved directly using
the monotone convergence theorem on
E
[
e
√
Z
I
{
Z
≥
x
}
].)
(e) We will now develop an upper bound for the second moment using importance
sampling. Show that
E
[
Y
2
]
≤
e

(
x

1)
2
/
2
e
3
.
(f) Using the above results, show that
var
(
X
)
var
(
Y
)
→ ∞
as
x
→ ∞
. (You should use L’Hopitals rule for computing the limit.) This
means that the improvement factor of the importance sampling algorithm over
the standard simulation algorithm tends to infinity as the event becomes rarer.
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 Spring '07
 sigman
 Derivative, Normal Distribution, Monotone convergence theorem

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