信度理论习题1以下几题来自近几年SOA的course4和courseC1、（2005 May course C 第2题）You are given:(i) The number of claims follows a negative binomial distribution with parameters r andβ＝3 .(ii) Claim severity has the following distribution:The number of claims is independent of the severity of claims.Determine the expected number of claims needed for aggregate losses to be within 10% of expected aggregate losses with 95% probability.2、2005FallcourseC 第35题. You are given:(i) The number of claims follows a Poisson distribution.(ii) Claim sizes follow a gamma distribution with parameters α (unknown) and 10,000(iii) The number of claims and claim sizes are independent.(iv) The full credibility standard has been selected so that actual aggregate losses will be within 10% of expected aggregate losses 95% of the time.Using limited fluctuation (classical) credibility, determine the expected number of claims required for full credibility.((A) Less than 400(B) At least 400, but less than 450(C) At least 450, but less than 500(D) At least 500(E) The expected number of claims required for full credibility cannot be determined from the information given.这道题limited fluctuation (classical) credibility就是指我们课堂上讲的有限波动信度，这道题要求我们求满足完全可信条件所需的最小理赔次数。
3、1104－第21题 You are given:(i) The number of claims has probability function:(ii) The actual number of claims must be within 1% of the expected number of claims with probability 0.95.(iii) The expected number of claims for full credibility is 34,574.Determine q.4、1100中第14题. For an insurance portfolio, you are given:(i) For each individual insured, the number of claims follows a Poisson distribution.(ii) The mean claim count varies by insured, and the distribution of mean claim countsfollows a gamma distribution.(iii) For a random sample of 1000 insureds, the observed claim counts are as follows:(iv) Claim sizes follow a Pareto distribution with mean 1500 and variance 6,750,000.(v) Claim sizes and claim counts are independent.(vi) The full credibility standard is to be within 5% of the expected aggregate loss 95% of the time.Determine the minimum number of insureds needed for the aggregate loss to be fully credible.提示：这道题中考到两个知识点：混合分布和完全可信条件。
做题时大家要考虑一下题目中给的表是用来计算什么的？信度理论习题21、1104-13. You are given:(i) The number of claims observed in a 1-year period has a Poisson distribution with mean θ.(ii) The prior density（先验密度） is:iii) The unconditional probability of observing zero claims in 1 year is 0.575.Determine k.2、1104－33. 33. You are given:(i) In a portfolio of risks, each policyholder can have at most one claim per year.(ii) The probability of a claim for a policyholder during a year is q .(iii) The prior density is ,A randomly selected policyholder has one claim in Year 1 and zero claims in Year 2.For this policyholder, determine the posterior probability(后验概率) that 3、1100－11For a risk, you are given:(i) The number of claims during a single year follows a Bernoulli distribution with mean p.(ii) The prior distribution for p is uniform on the interval [0,1].(iii) The claims experience is observed for a number of years.(iv) The Bayesian premium is calculated as 1/5 based on the observed claims.Which of the following observed claims data could have yielded this calculation?(A) 0 claims during 3 years(B) 0 claims during 4 years(C) 0 claims during 5 years(D) 1 claim during 4 years(E) 1 claim during 5 years4、1100－23 You are given:(i) The parameter L has an inverse gamma distribution with probability density function:(ii) The size of a claim has an exponential distribution with probability density function:For a single insured, two claims were observed that totaled 50.Determine the expected value of the next claim from the same insured.5、1100－28. Prior to observing any claims, you believed that claim sizes followed a Pareto distribution with parameters q =10 and, with each value being equally likely.You then observe one claim of 20 for a randomly selected risk.Determine the posterior probability that the next claim for this risk will be greater than 30.6、1103－7. You are given:(i) The annual number of claims for a policyholder has a binomial distribution with probability function: (ii) The prior distribution is:This policyholder had one claim in each of Years 1 and 2.Determine the Bayesian estimate of the number of claims in Year 3.信度理论习题31、1104-25. You are given:(i) A portfolio of independent risks is divided into two classes.(ii) Each class contains the same number of risks.(iii) For each risk in Class 1, the number of claims per year follows a Poisson distribution with mean 5.(iv) For each risk in Class 2, the number of claims per year follows a binomial distribution with m=8and q = 0.55(v) A randomly selected risk has three claims in Year 1, r claims in Year 2 and four claims in Year 3.The Bühlmann credibility estimate f。