CS1A Specimen Exam Paper 2019 - Institute and Faculty of Actuaries

Q: How many questions must be attempted and how should answers be presented?

All 11 questions must be attempted, starting each answer on a new page. Calculations should be shown where appropriate, and graph paper is not required.

Q: What does the paper mean by censored data in Question 1?

Some sample times are marked with a plus sign, indicating the policies had not yet produced a claim at the time of the survey, so those times are treated as censored observations.

Q: Which topics are covered across the questions?

The paper covers median and conclusions about the mean from censored data, moment generating functions, distributions from joint probability tables, sampling distributions for normal models and the t-distribution, Bayesian estimation with conjugate priors, and GLM/log-link style modelling for Poisson and canonical-link exponential-family forms for the gamma distribution.

Sep 27, 2026

Uploaded by aboduw

CS1A Actuarial Statistics Curriculum 2019 specimen examination by the Institute and Faculty of Actuaries, with a 3 hours 15 minutes time allowance. The paper provides candidate instructions and then 11 actuarial statistics questions covering censored survival-style data, median and mean behavior, moment generating functions and their use to derive mean and variance, joint distributions and independence, derivation of a risk index distribution, sampling from a normal model with distributions of X̄ and S², the t-distribution’s relationship, Bayesian updating with Normal and gamma priors, Poisson GLM link functions, and exponential-family form for the gamma distribution with canonical links.

LanguageEnglish

CategoryExam

FormatPDF

Pages7

Reading Time18 min

AccessFree to download

Instructions to the candidate

AT THE END OF THE EXAMINATION

Questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

INSTITUTE AND FACULTY OF ACTUARIES Curriculum 2019 SPECIMEN EXAMINATION Subject CS1A – Actuarial Statistics Time allowed: Three hours and fifteen minutes INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate and examination details as requested on the front of your answer booklet. 2. You must not start writing your answers in the booklet until instructed to do so by the supervisor. 3. Mark allocations are shown in brackets. 4. Attempt all 11 questions, beginning your answer to each question on a new page. 5. Candidates should show calculations where this is appropriate. Graph paper is NOT required for this paper. AT THE END OF THE EXAMINATION Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this question paper. In addition to this paper you should have available the 2002 edition of the Formulae and Tables and your own electronic calculator from the approved list. CS1A Specimen Exam Paper 2019 􀂤 Institute and Faculty of Actuaries 1 (CT3 September 2010) In a survey, a sample of 10 policies is selected from the records of an insurance company. The following data give, in ascending order, the time (in days) from the start date of the policy until a claim has arisen from each of the policies in the sample. 297 301 312 317 355 379 404 419 432+ 463+ Some of the policies have not yet resulted in any claims at the time of the survey, so the times until they each give rise to a claim are said to be censored. These values are represented with a plus sign in the above data. (i) Calculate the median of this sample. [2] (ii) State what you can conclude about the mean time until claims arise from the policies in this sample. [2] [Total 4] 2 (CT3 April 2014) Let Xbe a random variable with probability density function: 􀂭 1 􀂰 2 ex ; x ≤ 0 f (x) = 􀂮 􀂰􀂯 ~~ ~~12 e−x ; x > 0 (i) Show that the moment generating function of X is given by: MX (t) = (1 − t 2 )−1 , for t < 1 . [3] (ii) Hence find the mean and the variance of X using the moment generating function in part (i) . [3] [Total 6] 3 (CT3 September 2011) Consider the random variable X taking the value X= 1 if a randomly selected person is a smoker, or X= 0 otherwise. The random variable Y describes the amount of physical exercise per week for this randomly selected person. It can take the values 0 (less than one hour of exercise per week), 1 (one to two hours) and 2 (more than two hours of exercise per week) . The random variable R = (3 − Y)2(X+ 1) is used as a risk index for a particular heart disease. The joint distribution of X and Y is given by the joint probability function in the following table. X 0 Y 1 2 0 1 0.2 0.1 0.3 0.1 0.25 0.05 (i) Calculate the probability that a randomly selected person does more than two hours of exercise per week. [1] (ii) Decide whether X and Y are independent or not and justify your answer. [2] (iii) Derive the probability function of R. [3][Total 6] 4 (CT3 April 2013) Consider a random sample, X1 , , Xn, from a normal N(µ, σ2) distribution, with sample mean X and sample variance S 2 . (i) Define carefully what it means to say that X1 ,…, Xn is a random sample from a normal distribution. [2] (ii) State what is known about the distributions of X and S 2 in this case, including the dependencies between the two statistics. [3] (iii) Define the t-distribution and explain its relationship with X and S 2 . [2][Total 7] 5 (CT6 April 2014) The heights of adult males in a certain population are Normally distributed with unknown mean µ and standard deviation σ = 15. Prior beliefs about µ are described by a Normal distribution with mean 187 and standard deviation 10. (i) Calculate the prior probability that µ is greater than 180. [2] A sample of 80 men is taken and the mean height is found to be 182. (ii) Calculate the posterior probability that µ is greater than 180. [4] (iii) Comment on your results from pa

FAQs

How many questions must be attempted and how should answers be presented?

All 11 questions must be attempted, starting each answer on a new page. Calculations should be shown where appropriate, and graph paper is not required.

What does the paper mean by censored data in Question 1?

Some sample times are marked with a plus sign, indicating the policies had not yet produced a claim at the time of the survey, so those times are treated as censored observations.

Which topics are covered across the questions?

The paper covers median and conclusions about the mean from censored data, moment generating functions, distributions from joint probability tables, sampling distributions for normal models and the t-distribution, Bayesian estimation with conjugate priors, and GLM/log-link style modelling for Poisson and canonical-link exponential-family forms for the gamma distribution.

Sep 27, 2026

Uploaded by aboduw