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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.

356 Appendix A Random Variables and Probability Distributions whereW 1 isacontinuous random variable. Ifthedistribution of W 1 isexponential with parameter 1, then the distribution function of W is F(x) = 0, if x < , 1 2 + 1 2 1 −e −x = 1 − 1 2 e , if x ≥ 0. This distribution function is neither continuous (since it has a discontinuity at x = 0) nor discrete (since it increases ...

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