individual risk model
Claims occur on a portfolio of 100 general insurance policies according to a Poisson process. The expected number of claims per annum on each policy is , and the claim size distribution has density function f(x), where 1 f(x) = xe-x/100 x > 0 10000 The parameter is not the same for all policies in the portfolio, but is modelled as a random variable (independent of the claim sizes) with density function g(2), where: g(1) = 1001e-102 2> 0 (i) Calculate the mean and variance of annual aggregate claims. (ii) Given an initial reserve of 2,000, use a normal approximation to the distribution of aggregate claims to find the relative premium loading that should be used in order to be 95% sure that the reserve at the end of the first year is positive. (iii) If were fixed at its mean value, describe the effect that this would have on the value of the relative premium loading. (No further calculations are required.)
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A portfolio of car insurance claims has claim amounts following the distribution 7 x 3307 f(x)(33...
A portfolio of car insurance claims has claim amounts following the distribution 7 x 3307 f(x)(330+x The number of claims from this portfolio has a Poisson distribution with a mean of 500 per year. The premium income from this portfolio is a total of £30 000 per year. The insurer has an initial surplus of £2000. (a) Find the security loading factor for this example. (b) By assuming that the aggregate claims distribution is approximately nomal what is the probability...
Question 3-6 An insurance company each policy follows a Poisson distribution with a mean 3. has issued 75 policies. The number of claims filed under Assuming that the claims filed by each policyh...
Question 3-6 An insurance company each policy follows a Poisson distribution with a mean 3. has issued 75 policies. The number of claims filed under Assuming that the claims filed by each policyholder are independent of each other, what is the approximate probability that more than 250 claims will be filed by the group of policyholders? A) 0.048 B 0.168 C) 0.424 D) 0.576 E) 0.952 Question 3-7 650X and let X have the following probability density function: Let Y...
1. The size of claims made on an insurance policy are modelled through the following distribu- ti...
1. The size of claims made on an insurance policy are modelled through the following distribu- tion: You are interested in estimating the parameter λ > 0, using the following observations: 120, 20, 60, 70, 110, 150, 220, 160, 100, 100 (a) Verify that f is a density (b) Find the expectation of the generic random variable X, as a function of \ when A 1 (c) Prove that the method of moments estimator of λ is λι =斉. Calculate...
1. (3 POINTS) An insurance policy pays a individual $500 per day for up to 3...
1. (3 POINTS) An insurance policy pays a individual $500 per day for up to 3 days of hospitalization and $100 per day for each day of hospitalization thereafter. The number of days of hospitalization is a random variable X with P(X=x) = (6-x)/15, if x = 1,2,3,4,5. Calculate the expected payment for hospitalization under this policy. 2. (4 POINTS) An insurance policy reimburses a loss up to a benefit limit 0f $10. There is no deductible. The policyholder’s...
15 Consider a block of N-10000 risk units j. each distribution of claims of compound mixed Poisso...
15 Consider a block of N-10000 risk units j. each distribution of claims of compound mixed Poisson type baring a same expected number n,-0.1 of claims. The standard Ht ion of the mixing variable q, of each risk uni js o 02. re, the joint claim size d.f. is approximated by a discrete monetary unit, say £1000, in ntble, where s, is the probability that the size of a claim as given in terms of some suitable Z, 16 s,...
Reserve Problems Chapter 5 Section 7 Problem 4 The computational time of a statistical analysis applied...
Reserve Problems Chapter 5 Section 7 Problem 4 The computational time of a statistical analysis applied to a data set can sometimes increase with the square of N, the number of rows of data. Suppose that for a particular algorithm, the computation time is approximately T=0.004N seconds. Although the number of rows is a discrete measurement, assume that the distribution of N is over a number of data sets can be approximated with an exponential distribution with a mean of...
4. Suppose that N is a random variable having a conditional Poisson distribution with ability mass...
4. Suppose that N is a random variable having a conditional Poisson distribution with ability mass function prob- 1 (log 3) PN(i) i 1,2,3,... 2 i (a) Show that the mean of N is 3 log 3 1.6479, 2 and the variance of N is 3(log 3)2 3 log 3 0.7427. 2 4 (b) Calculate the probability P(N -4I 20). (c) Use the Bienaymé-Chebyshev inequality to give a lower bound for the probability that N takes values within 2 standard...
1. The size of claims made on an insurance policy are modelled through the following distribu- ti...
ANSWER QUESTION 2
1. The size of claims made on an insurance policy are modelled through the following distribu- tion: λ+1 You are interested in estimating the parameter λ > 0, using the following observations 120, 20, 60, 70, 110, 150, 220, 160, 100, 100 (a) Verify that f is a density (b) Find the expectation of the generic random variable X, as a function of when > 1 (c) Prove that the method of moments estimator of λ is...
solve d, e, f ASP please Let Y denote the claim size for a certain type...
solve d, e, f
ASP please
Let Y denote the claim size for a certain type of insurance policy and suppose Y has an exponential distribution with mean $300,000. (a) Write the density of Y for all values of y. (b) Determine the probability that a claim is smaller than $400,000 Define X = -Y so that X gives claim sizes that are negative, representing a payout from the insurance company. The remaining parts of this problem involve analyzing the...
Solve for the following probabilities (ranges of X values): a. P(X ≤ 7) when m = 15...
Solve for the following probabilities (ranges of X values): a. P(X ≤ 7) when m = 15 b. P(9 ≤ X ≤ 18) when m = 15 c. P(X ≥ 15) when m = 15 d. P(12 ≤ X < 20) when m = 15 (1) Solve the probabilities on the PROB worksheet. (To four decimal places) The P(X ≤7) when the poisson mean = 15 is ________.Assume a poisson distribution (2) Solve the probabilities on the PROB worksheet. (To four decimal places) The...
A portfolio of car insurance claims has claim amounts following the distribution 7 x 3307 f(x)(330+x The number of claims from this portfolio has a Poisson distribution with a mean of 500 per year. The premium income from this portfolio is a total of £30 000 per year. The insurer has an initial surplus of £2000. (a) Find the security loading factor for this example. (b) By assuming that the aggregate claims distribution is approximately nomal what is the probability...
Question 3-6 An insurance company each policy follows a Poisson distribution with a mean 3. has issued 75 policies. The number of claims filed under Assuming that the claims filed by each policyholder are independent of each other, what is the approximate probability that more than 250 claims will be filed by the group of policyholders? A) 0.048 B 0.168 C) 0.424 D) 0.576 E) 0.952 Question 3-7 650X and let X have the following probability density function: Let Y...
1. The size of claims made on an insurance policy are modelled through the following distribu- tion: You are interested in estimating the parameter λ > 0, using the following observations: 120, 20, 60, 70, 110, 150, 220, 160, 100, 100 (a) Verify that f is a density (b) Find the expectation of the generic random variable X, as a function of \ when A 1 (c) Prove that the method of moments estimator of λ is λι =斉. Calculate...
15 Consider a block of N-10000 risk units j. each distribution of claims of compound mixed Poisson type baring a same expected number n,-0.1 of claims. The standard Ht ion of the mixing variable q, of each risk uni js o 02. re, the joint claim size d.f. is approximated by a discrete monetary unit, say £1000, in ntble, where s, is the probability that the size of a claim as given in terms of some suitable Z, 16 s,...
Reserve Problems Chapter 5 Section 7 Problem 4 The computational time of a statistical analysis applied to a data set can sometimes increase with the square of N, the number of rows of data. Suppose that for a particular algorithm, the computation time is approximately T=0.004N seconds. Although the number of rows is a discrete measurement, assume that the distribution of N is over a number of data sets can be approximated with an exponential distribution with a mean of...
4. Suppose that N is a random variable having a conditional Poisson distribution with ability mass function prob- 1 (log 3) PN(i) i 1,2,3,... 2 i (a) Show that the mean of N is 3 log 3 1.6479, 2 and the variance of N is 3(log 3)2 3 log 3 0.7427. 2 4 (b) Calculate the probability P(N -4I 20). (c) Use the Bienaymé-Chebyshev inequality to give a lower bound for the probability that N takes values within 2 standard...
ANSWER QUESTION 2
1. The size of claims made on an insurance policy are modelled through the following distribu- tion: λ+1 You are interested in estimating the parameter λ > 0, using the following observations 120, 20, 60, 70, 110, 150, 220, 160, 100, 100 (a) Verify that f is a density (b) Find the expectation of the generic random variable X, as a function of when > 1 (c) Prove that the method of moments estimator of λ is...
solve d, e, f
ASP please
Let Y denote the claim size for a certain type of insurance policy and suppose Y has an exponential distribution with mean $300,000. (a) Write the density of Y for all values of y. (b) Determine the probability that a claim is smaller than $400,000 Define X = -Y so that X gives claim sizes that are negative, representing a payout from the insurance company. The remaining parts of this problem involve analyzing the...
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