Random Variables and Probability Distributions

ARandom Variablesand Probability Distribution Functions and Random The Multivariate Normal Distribution Functions and ExpectationThe distribution functionFof a Random variableXis defined byF(x)=P[X x]( )for all realx. The following properties are direct consequences of ( ) nondecreasing, ,F(x) F(y)ifx right continuous, ,F(y) F(x)asy (x) 1andF(y) 0asx andy , , any function that satisfies properties 1 3 is the distribution function ofsome Random of the commonly encountered distribution functionsFcan be expressedeither asF(x)= x f(y)dy( )orF(x)= j:xj xp(xj),( )where{x0,x1,x2,..}is a finite or countably infinite set. In the case ( )weshallsay that the Random variableXiscontinuous. The functionfis called theprobabilitydensity function(pdf) ofXand can be found from the relationf(x)=F (x). Springer International Publishing Switzerland Brockwell, Davis,Introduction to Time Series and Forecasting,Springer Texts in Statistics, DOI A Random Variables and Probability DistributionsIn case ( ), the possible values ofXare restricted to the set{x0,x1.}

A Random Variables and Probability Distributions A.1 Distribution Functions and Expectation A.2 Random Vectors A.3 The Multivariate Normal Distribution A.1 Distribution Functions and Expectation The distribution function F of a random variable X is defined by F(x) = P[X ≤ x] (A.1.1) for all real x. The following properties are direct ...

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