and so the posterior distribution is inverse gamma 2 f with and The Bayesian

And so the posterior distribution is inverse gamma 2

This preview shows page 110 - 112 out of 261 pages.

, and so the posterior distribution is inverse gamma 2 f&*)° + 0 0 with and . The Bayesian premium is the predictive mean. Since $ , 0 0 c c ( ~ * ~ &*) I´[ ` µ ~ ( is exponential given ), it follows that the predictive mean is equal to the posterior mean, [ 0 which is , $ c c f% *f% &*) ~ ~ )' À The variance of the predictive distribution is I´[ `B ~ ($Á B ~ )$Á B ~ +)µ f ²I´[ `B ~ ($Á B ~ )$Á B ~ +)µ³ ( & & % & ' ( % & ' . We have found the predictive mean to be . I´[ `B ~ ($Á B ~ )$Á B ~ +)µ ~ )' ( % & ' I´[ `B ~ ($Á B ~ )$Á B ~ +)µ ~ I´[ ` µ i ² `B ~ ($Á B ~ )$Á B ~ +)³ 1 ( ( & & % & ' % & ' $ ^ ( 0 4 0 0 . Since is exponential given , we have [ I´[ ` µ ~ & 0 0 0 ( & & (the second moment of an exponential distribution is 2 times the square of the mean). Therefore, I´[ `B ~ ($Á B ~ )$Á B ~ +)µ ~ & i ² `B ~ ($Á B ~ )$Á B ~ +)³ 1 ( & & % & ' % & ' $ ^ ( 0 4 0 0 , which is (2nd moment of the posterior distribution). Since the posterior is inverse gamma, its & g 2nd moment is . The predictive variance is ² ³ ²&*)³ ² f%³² f&³ ²)³²(³ , $ $ c & & c c ~ ~ 'Á )%%À&) &²'Á )%%À&)³ f ²)'³ ~ (Á &%'À&) & . ¡ Example CR-25: Tom has a coin, but he doesn't know the probability of tossing a head. He assumes a prior distribution for the probability of tossing a head to be = 4 ²=³ ~ &= $ ¡ = ¡ % for . Tom tosses the coin 10 times and observes 4 heads. Find the posterior distribution of the probability of tossing a head. Suppose that Tom tosses the coin another 5 times. Find the expected number of heads in those next 5 tosses. Solution: We first observe that , which is a beta 4 ²=³ ~ &= ~ i = ²% f =³ " " " ²'³ ²&³ ²%³ &f% %f% distribution with , . The number of heads tossed in 10 tosses has a binomial . ~ & / ~ % distribution with parameters and , and probability function ; ~ %$ = 3²B`=³ ~ = ²% f =³ B = 4 5 %$ B B %$fB . The joint density of and is 3²(Á =³ ~ 3²(`=³ i ²=³ ~ = ²% f =³ i i = ²% f =³ 4 4 5 %$ ²'³ ²&³ ²%³ ( ( %$f( &f% %f% " " " , which is proportional to . Therefore, the posterior is also a beta distribution, with and . = ²% f =³ . ~ * / ~ + *f% +f% c c The predictive mean for the number of heads in 5 more tosses is ` µ ~ `=µ i ²=` ³ 1= no. heads in 5 more tosses 4 heads no. heads in 5 more tosses 4 heads ( $ % 4 ~ )= i ²=` ³ 1= ~ ) g = ~ ) i ~ ) i ~ &À' ( $ % 4 4 heads (posterior mean of ) . . * . e/ *e+ c c c ¡
Image of page 110
CR-1 0 8 CREDIBILITY SECTION 5 - BAYESIAN CREDIBILITY, DISTRIBUTION TABLE © ACTEX 2007 SOA Exam C/CAS Exam 4 - Construction and Evaluation of Actuarial Models A variation on the Beta prior and Negative Binomial model distributions The parametrization of the negative binomial distribution in the Exam C table has parameters ? and , both . A variation on this parametrization is to keep the parameter , but use the % ¢ $ ? parameter , where so that . Suppose that we use this parametrization, and > > ~ $ ¡ > ¡ % % %e % that we assume that is a prior parameter with a beta distribution with parameters and , with > . / prior density . Assume that the conditional distribution of 4 ²>³ ~ i > ²% f >³ [ " " " ².e/³ ².³i ²/³ .f% /f% given is binomial with known parameter , and with parameter , so that the model distribution > : > has probability function for . 3²B`>³ ~ > ²% f >³ B ~ $Á %Á ÀÀÀÁ : 2 3 : B B :fB The joint distribution of and has joint density [ > 3²BÁ >³ ~ 3²B`>³ i ²>³ ~ > ²% f >³ i i > ²% f >³ 4 2 3 : B ².³i ²/³ ².e/³ B :fB .f% /f% " " " ~ i i > ²% f >³ 2 3 : B ².³i ²/³ ².e/³ " " " .eBf% /e:fBf% .
Image of page 111
Image of page 112

You've reached the end of your free preview.

Want to read all 261 pages?

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture