Hello!
I am trying to find out if the following integral
(given in mathematica notation) can be solved analytically.
If anyone knows the answer to this I'd appreciate the help.
f[y_]:=p*Exp[-y^2/(2*a)]/Sqrt[2*Pi*a] + (1-p)*Exp[-y^2/(2*b)]/Sqrt[2*Pi*b]
Integrate[-f[y]*Log[f[y]] , {y, -Infinity, Infinity}]
where 0 < p < 1 and a,b > 0.
******
Notice that this is the (shannon) entropy of a gaussian-mixed
random variable ,Y. In this case both components in the mixture
are gaussian themselves with zero mean and variances a and b,
and they are weighted according to p.
I am also interested in the case when one of the gaussian
components doesn't have a zero mean.
If anyone knows if this entropy can or can't be computed
analytically please let me know.
(please email me directly ,if possible, as well as post here)
Thanks,
Dave.
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