Chapter 1 Claim-Frequency Distribution

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Nonlife Actuarial ModelsChapter 1Claim- frequency DistributionLearning Objectives discrete distributions for modeling claim frequency Binomial, geometric, negative binomial and Poisson distributions The (a, b,0) and (a, b,1) classes of distributions Compound Distribution Convolution Mixture Review of Statistics Distribution function(df) of random variableXFX(x)=Pr(X x).( ) Probability density function(pdf) of continuous random vari-ablefX(x)=dFX(x)dx.( ) Probability function(pf) of discrete random variablefX(x)=(Pr(X=x),ifx X,0,otherwise.( )where Xis the support ofX3 Moment generating function(mgf), defined asMX(t)=E(etX).( ) Moments ofXare obtainable from mgf byMrX(t)=drMX(t)dtr=drdtrE(etX)=E(XretX) ,( )so thatMrX(0) = E(Xr)= 0r.( )4 IfX1,X2, ,Xnareindependently and identically distrib-uted (iid)random variables with mgfM(t), andX=X1+ +Xn,then the mgf ofXisMX(t)=E(etX)=E nYi=1etXi!)

• Discrete distributions for modeling claim frequency • Binomial, geometric, negative binomial and Poisson distributions • The (a,b,0) and (a,b,1) classes of distributions

  Claim, Discrete, Frequency, Claim frequency