November 2013
Intermediate to advanced
244 pages
6h 47m
English
book
Handbook of Differential Entropy
by Joseph Victor Michalowicz, Jonathan M. Nichols, Frank Bucholtz
Content preview from Handbook of Differential Entropy
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Differential Entropies for Probability Distributions 151
Using integration by parts with:
u = x − µ dv = (x − µ)e
−(x−µ)
2
/2σ
2
dx
du = dx v = −σ
2
e
−(x−µ)
2
/2σ
2
yields
h
X
= ln
Zσ
√
2π
+
1
2Zσ
3
√
2π
"
−σ
2
(x − µ)e
−(x−µ)
2
/2σ
2
|
b
a
+ σ
2
Z
b
a
e
−(x−µ)
2
/2σ
2
dx
#
= ln
Zσ
√
2π
+
1
2Z
√
2π
a − µ
σ
e
−(a−µ)
2
/2σ
2
−
b − µ
σ
e
−(b−µ)
2
/2σ
2
+
1
2
Z
b
a
1
Zσ
√
2π
e
−(x−µ)
2
/2σ
2
dx
= ln
Zσ
√
2π
+
1
2Z
a − µ
σ
φ
a − µ
σ
−
b − µ
σ
φ
b − µ
σ
+
1
2
= ln
σZ
√
2πee
1
2Z
[(
a−µ
σ
)
φ
(
a−µ
σ
)
−
(
b−µ
σ
)
φ
(
b−µ
σ
)]
In bits, the differential entropy becomes
h
X
= log
2
√
2πeσZe
1
2Z
[(
a−µ
σ
)
φ
(
a−µ
σ
)
−
(
b−µ
σ
)
φ
(
b−µ
σ
)]
152 Handbook of Differential Entropy
4.27 Uniform ...
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