Revision history [back]
 | 1 | initial version |
I don't know if this is what you expected but maxima can integrate the entropy
sage:maxima_console()
(%i1) dispay2d:false;
(%o1) false
(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$
(%i3) assume(sigma>0)$
(%i4) assume(mu>0)$
(%i5) ii:integrate(-q*log(q),x,minf,inf)$
(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$
(%i7) ev(%,integrate);
5/2 3/2 3/2
(%o7) (sqrt(%pi) (2 sigma log(sigma) + (2 log(%pi) + 2 log(2)) sigma)
3/2 5/2
+ 2 sqrt(%pi) sigma)/(2 sqrt(%pi) sigma)
| | 2 | No.2 Revision |
I don't know if this is what you expected but maxima can integrate the entropy
sage:maxima_console()
(%i1) dispay2d:false;
(%o1) false
(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$
(%i3) assume(sigma>0)$
(%i4) assume(mu>0)$
(%i5) ii:integrate(-q*log(q),x,minf,inf)$
(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$
(%i7) ev(%,integrate);
5/2 3/2 3/2
(%o7) (sqrt(%pi) (2 sigma log(sigma) + (2 log(%pi) + 2 log(2)) sigma)
3/2 5/2
+ 2 sqrt(%pi) sigma)/(2 sqrt(%pi) sigma)
If ex is sageized version of the last expression then
sage: ex.full_simplify()
1/2*log(pi) + 1/2*log(2) + log(sigma) + 1/2
| | 3 | No.3 Revision |
I don't know if this is what you expected but maxima can integrate compute the entropy
sage:maxima_console()
(%i1) dispay2d:false;
(%o1) false
(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$
(%i3) assume(sigma>0)$
(%i4) assume(mu>0)$
(%i5) ii:integrate(-q*log(q),x,minf,inf)$
(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$
(%i7) ev(%,integrate);
5/2 3/2 3/2
(%o7) (sqrt(%pi) (2 sigma log(sigma) + (2 log(%pi) + 2 log(2)) sigma)
3/2 5/2
+ 2 sqrt(%pi) sigma)/(2 sqrt(%pi) sigma)
If ex is sageized version of the last expression then
sage: ex.full_simplify()
1/2*log(pi) + 1/2*log(2) + log(sigma) + 1/2
| | 4 | No.4 Revision |
I don't know if this is what you expected but maxima can compute the entropy
sage:maxima_console()
(%i1) dispay2d:false;
(%o1) false
(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$
(%i3) assume(sigma>0)$
(%i4) assume(mu>0)$
(%i5) ii:integrate(-q*log(q),x,minf,inf)$
(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$
(%i7) ev(%,integrate);
5/2 3/2 3/2
(%o7) (sqrt(%pi) (2 sigma log(sigma) + (2 log(%pi) + 2 log(2)) sigma)
3/2 5/2
+ 2 sqrt(%pi) sigma)/(2 sqrt(%pi) sigma)
If ex is sageized version of the last expression then
sage: ex.full_simplify()
1/2*log(pi) + 1/2*log(2) + log(sigma) + 1/2
and the last expression is equal: log(sqrt(2pi)sigma)+1/2
| | 5 | No.5 Revision |
I don't know if this is what you expected but maxima can compute the entropy
sage:maxima_console()
(%i1) dispay2d:false;
(%o1) false
(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$
(%i3) assume(sigma>0)$
(%i4) assume(mu>0)$
(%i5) ii:integrate(-q*log(q),x,minf,inf)$
(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$
(%i7) ev(%,integrate);
5/2 3/2 3/2
(%o7) (sqrt(%pi) (2 sigma log(sigma) + (2 log(%pi) + 2 log(2)) sigma)
3/2 5/2
+ 2 sqrt(%pi) sigma)/(2 sqrt(%pi) sigma)
If ex is sageized version of the last expression then
sage: ex.full_simplify()
1/2*log(pi) + 1/2*log(2) + log(sigma) + 1/2
and the last expression is equal: log(sqrt(2pi)sigma)+1/2 log(sqrt(2*pi)*sigma)+1/2
| | 6 | No.6 Revision |
I don't know if this is what you expected but maxima can compute the entropy
sage:maxima_console()
(%i1) dispay2d:false;
(%o1) false
(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$
(%i3) assume(sigma>0)$
(%i4) assume(mu>0)$
(%i5) ii:integrate(-q*log(q),x,minf,inf)$
(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$
(%i7) ev(%,integrate);
5/2 ev(%,integrate)$
(%i8) ratsimp(%);
3/2 2 log(sigma) + log(%pi) + log(2) + 1
(%o8) 3/2
(%o7) (sqrt(%pi) (2 sigma log(sigma) + (2 log(%pi) + 2 log(2)) sigma)
------------------------------------
3/2 5/2
+ 2 sqrt(%pi) sigma)/(2 sqrt(%pi) sigma)
If ex is sageized version of the last expression then
sage: ex.full_simplify()
1/2*log(pi) + 1/2*log(2) + log(sigma) + 1/2
2
and the last expression is equal: log(sqrt(2*pi)*sigma)+1/2
| | 7 | No.7 Revision |
I don't know if this is what you expected but maxima can compute the entropy
sage:maxima_console()
(%i1) dispay2d:false;
(%o1) false
(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$
(%i3) assume(sigma>0)$
(%i4) assume(mu>0)$
(%i5) ii:integrate(-q*log(q),x,minf,inf)$
(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$
(%i7) ev(%,integrate)$
(%i8) ratsimp(%);
2 log(sigma) + log(%pi) + log(2) + 1
(%o8) ------------------------------------
2
(%i9) logcontract(%);
2
log(2 %pi sigma ) + 1
(%o9) ---------------------
2
and the last expression is equal: log(sqrt(2*pi)*sigma)+1/2
| | 8 | No.8 Revision |
I don't know if this is what you expected but maxima Maxima can compute the entropy
sage:maxima_console()
(%i1) dispay2d:false;
(%o1) false
(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$
(%i3) assume(sigma>0)$
(%i4) assume(mu>0)$
(%i5) ii:integrate(-q*log(q),x,minf,inf)$
(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$
(%i7) ev(%,integrate)$
(%i8) ratsimp(%);
2 log(sigma) + log(%pi) + log(2) + 1
(%o8) ------------------------------------
display2d:false$ 2
(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$
(%i3) assume(sigma>0)$
(%i4) assume(mu>0)$
(%i5) ii:integrate(-q*log(q),x,minf,inf)$
(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$
(%i7) ev(%,integrate)$
(%i8) ratsimp(%);
(%o8) (2*log(sigma)+log(%pi)+log(2)+1)/2
(%i9) logcontract(%);
2
log(2 %pi sigma ) + 1
(%o9) ---------------------
2
(log(2*%pi*sigma^2)+1)/2
and the last expression is equal: equal to:
log(sqrt(2*pi)*sigma)+1/2
