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This detailed introduction to distribution theory uses no measure theorymaking it suitable for students in statistics and econometrics as well as forresearchers who use statistical methods Good backgrounds in calculusand linear algebra are important and a course in elementary mathematicalanalysis is useful but not required
A continuous random variable X is said to have exponential distribution with parameter If the pdf of X is (with >0) f(x) = ( e?x;x>0 0; otherwise (*) Remarks Very often the independent variable will be time t rather than x The exponential distribution is the special case of the gamma distribution with = 1 and = 1
2 ProbabilityDistributionFunctions Probability*distribution*function ( pdf ): Functionformappingrandomvariablestorealnumbers Discrete*randomvariable:
Stable distributions have threeexclusiveproperties which can be brie?ysummarized stating that they 1) areinvariant under addition 2) possess theirowndomain of attraction and 3) admit acanonical characteristic function In the following sections let us illustrate the above properties whichproviding necessary and su?cient conditions can be
Distribution theory is concerned with probability distributions of random variables with the emphasis on the types of random variables frequently used in the theory and application of statistical methods For instance in a statistical estimation problem we may need to
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In particular, the problem of deriving properties of probability distributions of statistics, such as the sample mean or sample standard deviation, based on assumptions on the distributions of the underlying random variables, receives much emphasis in distribution theory.
Thedistribution functionof the distribu- tion ofXor, simply, the distribution function ofX, is the functionF?F X:R? [0,1] given by F(x) = Pr(X?x),??< x