Reef Fish
2006-07-20 04:49:46 UTC
Several posters in the Gaussian mixture problem seemed to have
confused a MIXTURE with a LINEAR COMBINATION.
A linear combination of two Gaussian distributions is Gaussian.
(Discounting the pathological examples discussed in another thread).
A MIXTURE of two Gaussian distributions with the same variance
and different means is NOT Gaussian. Whether the distribution of
the MIXTURE is bi-modal or unimodal depends on how far apart the
two means are!
If you MIX two Gaussian distributions with the same mean but
different variances, you'll get a unimodal non-Gaussian distribution
that is Leptokurtic, while a unimodal mixture of two Gaussian
distributions with same variance and different means will be
Platykurtic.
Leptokurtosis and Platykurtosis refer to the kurtosis of the
distribution compared to that of a Gaussian distribution.
The preceding three paragraphs are my recollections from the
textbook (by Chester I. Bliss) I had jettisoned into the trash
can 39 years ago. But what I recalled about the mixture of
two Gaussian distributions and how they relate to the kurtosis
of a Gaussian distribution seemed not to be found in other
textbooks of statistics. That is why I recalled that factoid
while the rest of the book was obsolete even 39 years ago.
Okay everyone. Make this a simple exercise (which was not
given in the book).
Let X and Y both be Gaussian distributions with variance 1,
and means at a and b respectively.
Characterize the unimodality and bi-modality of the EQUAL
mixture of those two distributions in terms of a and b.
-- Reef Fish Bob.
confused a MIXTURE with a LINEAR COMBINATION.
A linear combination of two Gaussian distributions is Gaussian.
(Discounting the pathological examples discussed in another thread).
A MIXTURE of two Gaussian distributions with the same variance
and different means is NOT Gaussian. Whether the distribution of
the MIXTURE is bi-modal or unimodal depends on how far apart the
two means are!
If you MIX two Gaussian distributions with the same mean but
different variances, you'll get a unimodal non-Gaussian distribution
that is Leptokurtic, while a unimodal mixture of two Gaussian
distributions with same variance and different means will be
Platykurtic.
Leptokurtosis and Platykurtosis refer to the kurtosis of the
distribution compared to that of a Gaussian distribution.
The preceding three paragraphs are my recollections from the
textbook (by Chester I. Bliss) I had jettisoned into the trash
can 39 years ago. But what I recalled about the mixture of
two Gaussian distributions and how they relate to the kurtosis
of a Gaussian distribution seemed not to be found in other
textbooks of statistics. That is why I recalled that factoid
while the rest of the book was obsolete even 39 years ago.
Okay everyone. Make this a simple exercise (which was not
given in the book).
Let X and Y both be Gaussian distributions with variance 1,
and means at a and b respectively.
Characterize the unimodality and bi-modality of the EQUAL
mixture of those two distributions in terms of a and b.
-- Reef Fish Bob.