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An insurance company has an initial surplus of 100 and premium loading factor of 20%. Assume that claims arrive according to a Poisson process with parameter λ = 5 and the size of claims X₁ are iid random variables with X; ~ exp(). The time unit is 1 week. Assume that 1 month is 4 weeks. (a) Calculate the average number of claims on any given day, week and month. Let t'> 0 be an instance of time. Calculate the probability that at least one claim occurs within 5 days after t'. Calculate also the probability that at least 2 claims occur within 5 days after t'. (b) Let t = 2 months. Calculate the mean and variance of S(t) and of U(t), where (S(r)) r2o and (U(r))r≥o are the aggregate claim process and the surplus process of the described model.

An insurance company has an initial surplus of 100 and premium loading factor of 20%. Assume that claims arrive according to a Poisson process with parameter λ = 5 and the size of claims X₁ are iid random variables with X; ~ exp(). The time unit is 1 week. Assume that 1 month is 4 weeks. (a) Calculate the average number of claims on any given day, week and month. Let t'> 0 be an instance of time. Calculate the probability that at least one claim occurs within 5 days after t'. Calculate also the probability that at least 2 claims occur within 5 days after t'. (b) Let t = 2 months. Calculate the mean and variance of S(t) and of U(t), where (S(r)) r2o and (U(r))r≥o are the aggregate claim process and the surplus process of the described model.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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An insurance company has an initial surplus of 100 and premium loading factor of 20%. Assume that claims arrive according to a Poisson process with parameter \ 5 and the size of claims Xį are iid random variables with Xi exp(). The time unit is 1 week. Assume that 1 month = is 4 weeks. (a) Calculate the average number of claims on any given day, week and month. Let t'> 0 be an instance of time. Calculate the probability that at least one claim occurs within 5 days after t'. Calculate also the probability that at least 2 claims occur within 5 days after t'. (b) Let t = 2 months. Calculate the mean and variance of S(t) and of U(t), where (S(r))ro and (U(r))rzo are the aggregate claim process and the surplus process of the described model.
Transcribed Image Text:An insurance company has an initial surplus of 100 and premium loading factor of 20%. Assume that claims arrive according to a Poisson process with parameter \ 5 and the size of claims Xį are iid random variables with Xi exp(). The time unit is 1 week. Assume that 1 month = is 4 weeks. (a) Calculate the average number of claims on any given day, week and month. Let t'> 0 be an instance of time. Calculate the probability that at least one claim occurs within 5 days after t'. Calculate also the probability that at least 2 claims occur within 5 days after t'. (b) Let t = 2 months. Calculate the mean and variance of S(t) and of U(t), where (S(r))ro and (U(r))rzo are the aggregate claim process and the surplus process of the described model.
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