SOA EXAM STAM Detailed Solutions 2024/2025 Exam Questions and Correct Answers | New Update

SOA EXAM STAM Detailed Solutions
2024/2025 Exam Questions and Correct
Answers | New Update


SOA STAM 60

Calculate E[X5∣S]E[X5∣S] using Bayesian analysis. - 🧠ANSWER ✔✔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.




Page 1 of 110
COPYRIGHT©JOSHCLAY 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER:
619652435. TERMS OF USE. PRIVACY STATEMENT. ALL RIGHTS RESERVED

,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.
Page 2 of 110
COPYRIGHT©JOSHCLAY 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER:
619652435. TERMS OF USE. PRIVACY STATEMENT. ALL RIGHTS RESERVED

,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.
Page 3 of 110
COPYRIGHT©JOSHCLAY 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER:
619652435. TERMS OF USE. PRIVACY STATEMENT. ALL RIGHTS RESERVED

, 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. - 🧠ANSWER

✔✔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.



Page 4 of 110
COPYRIGHT©JOSHCLAY 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER:
619652435. TERMS OF USE. PRIVACY STATEMENT. ALL RIGHTS RESERVED