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
Content preview from Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
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9.6 LOCATION-SCALE FAMILIES
The pdf of a location-scale family of random variables maintains its basic shape when its location or scale parameters are varied.
Definition: Location-Scale Family Let X be a random variable with cdf fX(x). A location family is obtained via the transformation Y = X+u with 
. Its distribution has the form
(9.84) 
A scale family is obtained via the transformation Y = vX, where v>0. Its distribution has the form
(9.85) 
Combining the transformations to generate Y = u+vX results in a location-scale family, whose distribution has the form
(9.86) 
and pdf
TABLE 9.3 Minimal Sufficient Statistics for Location-Scale Families
| Statistic T | Distribution Family {Parameter  } |
|
Cauchy  , Laplace , logistic
|
| X(1) | Shifted exponential , uniform {a, known b} |
| Shifted ... |
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