November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
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4.14 MIXTURES
Let 
be the pdfs of random variables {xn} for 
.
Definition: Probability Mixture Random variable Y is a probability mixture if it can be expressed as the following weighted sum:
where 
and 
. It is also called a finite mixture.
A mixture is not a sum of the random variables like the sample mean. The sum in (4.250) is a convex combination because the coefficients are (i) nonnegative and (ii) they sum to 1. These two properties ensure that FY(y) is a valid pdf: 
and
(4.251) 
Note that different pdfs can be used in (4.250). If the pdfs turn out to be from the same family, differing only by the values of the distribution parameters, then the probability mixture is known as a parametric mixture.
Definition: Parametric Mixture (Discrete) Random variable Y is a parametric mixture ...
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