SOA EXAM STAM Detailed Solutions
(Graded A)
SOA STAM 60
Calculate E[X5∣S]E[X5∣S] using Bayesian analysis. CORRECT ANSWERS Bayesian
Credibility
Main idea: use the experience S to update our probabilities of heads.
Goal: Expected value of 5th toss, given the experience.
Step 1- Use the experience to calculate the likelihood of the goal, given the data.
P(data | coin)
ex) P(data|C1-4) = 0.5^4= 0.0625.
The probability of S occurring will vary depending on which coin was selected and
tossed. The probability of heads is consistent between tosses of the same coin.
Step 2- Identify the prior probabilities
Since our data depends on which coin was selected, our prior distribution is the
probability of selecting the coin.
ex) P(c1-4) = 4/6
Step 3) Multiply prior*likelihood and sum to get Probability of S (Law of Total Probability)
P(S) = P(S and c1-4)+P(s and c5)+p(s and c6)
Note that we are only given model dist S|coin and coin, so we use bayes theorem to
rearrange and solve for the joint probability for the law of total probability formula.
P(S)= 0.06120
Step 4 - Calculate the posterior probabilities
the idea here is to use S to update our probability of coin.
We use the experience to update prior to posterior!
P(coin 1-4 | S) = [(PS|c1-4)*P(c1-4)]/P(s)
,keep in mind that posterior probabilities are used in weighted average and should sum
to 1
Step 5 - Get the means to use to calculate predictive mean
*This part is where I mess up usually*
So, our goal is E(X|S). We only have info E(X|coin). This will be used in weighted
average multiplied by associated posterior probability.
Ex) E(X|c1-4) = 0.5(1)+0.5(0)=0.5.
This is because the coin toss can either take on value of 1 or 0 for number of heads.
Step 6 - Predictive Mean
E(X|S) = 0.5*.6808+...
= .5638
SOA STAM 29
Each risk has at most one claim each year
One randomly chosen risk has three claims during Years 1-6.
Calculate the posterior probability of a claim for this risk in Year 7. CORRECT
ANSWERS Bayesian Credibility
Goal- use the data to update the probability of claims for a given selected risk.
Step 1- Use the experience to calculate the likelihood.
data= 3 claims yrs 1-6
since there are two possible outcomes (claim or no claim), and 6 years observed (max 6
claims 6 years), use binomial dist with the annual claim probability given risk class.
P(3|class 1) = (6 ncr 3) (0.1)^3(1-0.1)^3 = 0.01458
Step 2 - Identify the prior probabilities
Since our model distribution is X|class, our prior distribution is p(class). prior
probabilities are given
Step 3- Multiply likelihood and prior and sum to get unconditional probability of the data
(law of total probability)
P(data) = 0.054238
Step 4- Calculate the posterior probabilities
,What we have:
model dist p(data|class)
prior dist
p(class)
to update prior to posterior, we condition on the experience that has occurred.
p(class|data) = p(data|class)*p(class)/p(data)
AHHH OK ITS MAKING SENSE!!
WE ARE CONDITIONING!!
BECAUSE WE KNOW MORE!! :)
P(Class 1 | data) = (0.7)(0.01458) / .054238 = .18817
Step 5 - Calculate the posterior probability of 1 CLAIM
remember, claim can be 0 or 1. We use p(x=1|class) and multiply by our updated class
probability (posterior)
posterior prob is 0.28313
SOA STAM 260
You are given:
Claim sizes follow an exponential distribution with mean θθ.
For 80% of the policies, θ=8θ=8.
For 20% of the policies, θ=2θ=2.
A randomly selected policy had one claim in Year 1 of size 5.
Calculate the Bayesian expected claim size for this policy in Year 2. CORRECT
ANSWERS Bayesian Credibility
Step 1- Likelihood using experience
f(x=5|theta)
Step 2- Multiply by prior
f(x=5|theta)*pi(theta)
(used for numerators of posterior)
Step 3- Sum these products to get unconditional probability of data
P(x=5)
, Step 4- Calculate posterior probabilities using the updated "After the fact" observations
P(theta=8 | data) = .8670
Step 5- Calculate hypothetical means
According to Coaching Actuaries, "The Hypothetical Means are the expected claim size
for different values of theta."
E(x2|theta=8) = theta = 8
Step 5 - Predicted expected value of next claim size for different values of theta given
the prior experien
answer is 7.2
SOA STAM 53
Calculate the variance of the aggregate loss. CORRECT ANSWERS Aggregate Loss
Models
The idea here is to look at all possible claim combinations and creating new
probabilities of these actual combinations occurring. the new probabilities should sum to
1.
According to Coaching Actuaries, the probability for 250 includes (2 ncr 1) because
there are two permutations of 250 and 50 (250, 50 or 50,250).
possibilities are 0,25,100,150,250,400
ES = 105
VS = 8100
SOA STAM 79
Losses come from a mixture of an exponential distribution with mean 100 and with
probability pp and an exponential distribution with mean 10,000 with probability 1−p1−p.
Losses of 100 and 2,000 are observed.
Determine the likelihood function of pp. CORRECT ANSWERS Maximum Likelihood
Estimators
This is somewhat conceptual.
(Graded A)
SOA STAM 60
Calculate E[X5∣S]E[X5∣S] using Bayesian analysis. CORRECT ANSWERS Bayesian
Credibility
Main idea: use the experience S to update our probabilities of heads.
Goal: Expected value of 5th toss, given the experience.
Step 1- Use the experience to calculate the likelihood of the goal, given the data.
P(data | coin)
ex) P(data|C1-4) = 0.5^4= 0.0625.
The probability of S occurring will vary depending on which coin was selected and
tossed. The probability of heads is consistent between tosses of the same coin.
Step 2- Identify the prior probabilities
Since our data depends on which coin was selected, our prior distribution is the
probability of selecting the coin.
ex) P(c1-4) = 4/6
Step 3) Multiply prior*likelihood and sum to get Probability of S (Law of Total Probability)
P(S) = P(S and c1-4)+P(s and c5)+p(s and c6)
Note that we are only given model dist S|coin and coin, so we use bayes theorem to
rearrange and solve for the joint probability for the law of total probability formula.
P(S)= 0.06120
Step 4 - Calculate the posterior probabilities
the idea here is to use S to update our probability of coin.
We use the experience to update prior to posterior!
P(coin 1-4 | S) = [(PS|c1-4)*P(c1-4)]/P(s)
,keep in mind that posterior probabilities are used in weighted average and should sum
to 1
Step 5 - Get the means to use to calculate predictive mean
*This part is where I mess up usually*
So, our goal is E(X|S). We only have info E(X|coin). This will be used in weighted
average multiplied by associated posterior probability.
Ex) E(X|c1-4) = 0.5(1)+0.5(0)=0.5.
This is because the coin toss can either take on value of 1 or 0 for number of heads.
Step 6 - Predictive Mean
E(X|S) = 0.5*.6808+...
= .5638
SOA STAM 29
Each risk has at most one claim each year
One randomly chosen risk has three claims during Years 1-6.
Calculate the posterior probability of a claim for this risk in Year 7. CORRECT
ANSWERS Bayesian Credibility
Goal- use the data to update the probability of claims for a given selected risk.
Step 1- Use the experience to calculate the likelihood.
data= 3 claims yrs 1-6
since there are two possible outcomes (claim or no claim), and 6 years observed (max 6
claims 6 years), use binomial dist with the annual claim probability given risk class.
P(3|class 1) = (6 ncr 3) (0.1)^3(1-0.1)^3 = 0.01458
Step 2 - Identify the prior probabilities
Since our model distribution is X|class, our prior distribution is p(class). prior
probabilities are given
Step 3- Multiply likelihood and prior and sum to get unconditional probability of the data
(law of total probability)
P(data) = 0.054238
Step 4- Calculate the posterior probabilities
,What we have:
model dist p(data|class)
prior dist
p(class)
to update prior to posterior, we condition on the experience that has occurred.
p(class|data) = p(data|class)*p(class)/p(data)
AHHH OK ITS MAKING SENSE!!
WE ARE CONDITIONING!!
BECAUSE WE KNOW MORE!! :)
P(Class 1 | data) = (0.7)(0.01458) / .054238 = .18817
Step 5 - Calculate the posterior probability of 1 CLAIM
remember, claim can be 0 or 1. We use p(x=1|class) and multiply by our updated class
probability (posterior)
posterior prob is 0.28313
SOA STAM 260
You are given:
Claim sizes follow an exponential distribution with mean θθ.
For 80% of the policies, θ=8θ=8.
For 20% of the policies, θ=2θ=2.
A randomly selected policy had one claim in Year 1 of size 5.
Calculate the Bayesian expected claim size for this policy in Year 2. CORRECT
ANSWERS Bayesian Credibility
Step 1- Likelihood using experience
f(x=5|theta)
Step 2- Multiply by prior
f(x=5|theta)*pi(theta)
(used for numerators of posterior)
Step 3- Sum these products to get unconditional probability of data
P(x=5)
, Step 4- Calculate posterior probabilities using the updated "After the fact" observations
P(theta=8 | data) = .8670
Step 5- Calculate hypothetical means
According to Coaching Actuaries, "The Hypothetical Means are the expected claim size
for different values of theta."
E(x2|theta=8) = theta = 8
Step 5 - Predicted expected value of next claim size for different values of theta given
the prior experien
answer is 7.2
SOA STAM 53
Calculate the variance of the aggregate loss. CORRECT ANSWERS Aggregate Loss
Models
The idea here is to look at all possible claim combinations and creating new
probabilities of these actual combinations occurring. the new probabilities should sum to
1.
According to Coaching Actuaries, the probability for 250 includes (2 ncr 1) because
there are two permutations of 250 and 50 (250, 50 or 50,250).
possibilities are 0,25,100,150,250,400
ES = 105
VS = 8100
SOA STAM 79
Losses come from a mixture of an exponential distribution with mean 100 and with
probability pp and an exponential distribution with mean 10,000 with probability 1−p1−p.
Losses of 100 and 2,000 are observed.
Determine the likelihood function of pp. CORRECT ANSWERS Maximum Likelihood
Estimators
This is somewhat conceptual.
