Biomedical Engineering Reference
In-Depth Information
Letting
1
n
,
q
p
,
β
=
and
α
=
we obtain
2
e
−
jt
α/β
+
pe
jt
βα
)
ln(
p
β
lim
n
ln
φ
y
(
t
)
=
lim
α
→
.
α
2
→∞
0
Applying L'H ospital's Rule twice,
e
−
jt
α/β
+
e
jt
βα
t
2
p
t
2
2
p
t
2
2
.
0
−
jtp
β
jtp
β
=
−
−
β
lim
α
→
ln
φ
y
(
t
)
=
lim
α
→
=−
2
α
2
0
Consequently,
exp
lim
n
y
(
t
)
e
−
t
2
/
2
lim
n
→∞
φ
y
(
t
)
=
ln
φ
=
.
→∞
(
t
) in the above lemma is the characteristic function for a Gaussian RV
with zero mean and unit variance. Hence, for large
n
and
a
The limiting
φ
<
b
P
(
a
<
b
)
F
(
b
)
F
(
a
)
P
(
a
<
x
<
b
)
=
y
<
≈
−
,
(5.33)
where
γ
1
√
2
2
e
−
α
/
2
d
F
(
γ
)
=
α
=
1
−
Q
(
γ
)
(5.34)
π
−∞
is the standard Gaussian CDF,
a
np
√
npq
,
−
b
np
√
npq
,
−
a
=
b
=
(5.35)
and
Q
(
) is Marcum's Q function which is tabulated in Tables A.8 and A.9 of the Appendix.
Evaluation of the above integral as well as the Gaussian PDF are treated in Section 5.5.
·
Example 5.3.3.
Suppose x is a Bernoulli random variable with n
=
5000
and p
=
0
.
4
. Find
P
(
x
≤
2048)
.
Solution.
The solution involves approximating the Bernoulli CDF with the Gaussian CDF
since
npq
1200 and
b
=
=
1200
>
3. With
np
=
2000,
npq
=
(2048
−
2000)
/
34
.
641
=
1
.
39, we find from Table A.8 that
P
(
x
≤
2048)
≈
F
(1
.
39)
=
1
−
Q
(1
.
39)
=
0
.
91774
.
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