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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9.5 EXPONENTIAL FAMILIES
In this section, we consider exponential families of random variables for which a sufficient statistic of low dimension is easily found.
Definition: Exponential Family An exponential family of random variables with scalar parameter θ has a pdf that can be expressed in the following form:
where 
, 
, c(x), and d(x) are known functions. It is called a family of random variables because the form above remains the same as θ is varied. For N iid random variables {Xn}, (9.42) becomes
For a vector of M parameters 
and a single random variable, the pdf is
and for N iid random variables and M parameters, (9.44) becomes
(9.45) 
An exponential family of random variables should not be confused with the exponential distribution described in Chapter 3 (which ...
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