And so the posterior distribution is inverse gamma 2

, 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
I´ ` µ ~ I´ `=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
¡

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%
.