Handbook of Differential Entropy

208 Handbook of Differential Entropy

This occurs when

1

2

1

σ

√

2π

=

1

σ

√

2π

e

−(x−µ)

2

/2σ

2

1

2

= e

−(x−µ)

2

/2σ

2

−ln(2) = −

(x − µ)

2

2σ

2

(x − µ)

2

= 2σ

2

ln(2) = σ

2

ln(4)

x = µ ± σ

p

ln(4)

So the distribution becomes bi-modal exactly when µ − σ

p

ln(4) > 0 or µ >

σ

p

ln(4) = 1.1774σ.

The Characteristic Function is given by:

E[e

itx

] =

1

2

E[e

itx

] for Gaussian with mean µ

+

1

2

E[e

itx

] for Gaussian with mean − µ

=

1

2

h

e

iµt−

1

2

σ

2

t

2

+ e

−iµt−

1

2

σ

2

t

2

i

=

1

2

e

−σ

2

t

2

/2



e

iµt

+ e

−iµt



= e

−σ

2

t

2

/2

cos(µt).

7.4 Derivation of Function L(α) Used in Derivation for

Entropy of Mixed-Gaussian Distribution

An analytic expression for the integral in Eqn. (4.4) in the derivation of the

differential entropy for the Mixed-Gaussian ...