Changes

In the continuous case the discrete frequency function is replaced by a continuous probability density function, (p. d. f.) , f(x), and H may be written

In the continuous case the discrete frequency function is replaced by a continuous probability density function, (p. d. f.) , f(x), and H may be written

<math>  H = \int f(x) \log f(x) dx  .</math>

<math>  H = - \int f(x) \log f(x) dx  .</math>

A problem here is that H becomes uncertain because of the infinitesimally small dx. A way to evade the problem is to define H in such way that H equals the logarithm of the volume – in analogy to the cardinality n in the discrete case - covered by a uniform p. d. f. over the volume. Thus H may be defined as

A problem here is that H becomes uncertain because of the infinitesimally small dx. A way to evade the problem is to define H in such way that H equals the logarithm of the volume – in analogy to the cardinality n in the discrete case - covered by a uniform p. d. f. over the volume. Thus H may be defined as

<math> H = \int f(x) \log f(x) dx .</math>

<math> H = - \int f(x) \log f(x) dx .</math>

The “volume” covered by a Gaussian p. d. f. is proportional to the volume of the ellipsoid of concentration and equal to <math>\sqrt{2\pi e^n |M|} </math>, where M is the moment matrix of the Gaussian.

The “volume” covered by a Gaussian p. d. f. is proportional to the volume of the ellipsoid of concentration and equal to <math>\sqrt{2\pi e^n |M|} </math>, where M is the moment matrix of the Gaussian.