1. The number of dental claims for each insured in a calendar year is distributed as a Geometric distribution with variance of

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1 1. The number of dental claims for each insured in a calendar year is distributed as a Geometric distribution with variance of The amount of each claim is distributed as a Pareto distribution with 1000 and 4. Let S be the random variable representing the aggregate claims per insured in a calendar year. a. (8 points) Calculate the ES [ ] and Var[ S ]. Var N [ ] ( 1) (4)(1)( 7.315) ()(1) E[ S] E[ N] E[ X ] ( ) (.5) Var( S) E[ N] Var( X ) Var( N)( E[ X ]) (1000) (4) 1000 (4 1) () 4 1 (.5) (7.315) 1,31,500

2 b. (8 points) The Henry Insurance Company has 1000 independent dental policies which have no deductible or upper limits. Assuming the normal distribution, calculate the 95% confidence interval for the total aggregate claims for a calendar year for the 1000 policies. E[ Port] (1000)(750) 750,000 Var[ Port] (1000)(1,31,500) CI 750, (1000)(1,31,500) (678,99 ; 81, 008) c. (4 points) Henry Insurance Company decides to implement a franchise deductible of 50 for each claim. Determine the expected number of claims per policy during the calendar year. 4 L C L 1000 N.5 N N ( F(50)) (.5)

3 . (10 points) The number of claims under a hospital indemnity policy follows a Poisson distribution with a mean of 1. The amount of each claim is distributed as Amount of claim Probability Let S be the random variable representing the aggregate claims per insured in a calendar year. Calculate the net stop loss premium for a stop loss policy that covers aggregate losses in excess of 500. Premium E[ S] E[ S 500] E[ N] E[ X ] E[ S 500] (1) (0.)(1000) (0.3)(000) (0.5)(3000) (0)( f (0)) (1000)( f (1000)) (000)( f (000)) (500)(1 F (000)) S S S S f (0) Pr[ N 0] e S 1 f N X e 1 S (1000) Pr[ 1& 1000] ( )(0. ) e fs (000) Pr[ N 1& X 000] Pr[ N & Xs 1000] ( e )(0.3) (0.) e (0)( e ) (1000)( e 0.) (000) ( e )(0.3) (0.) (500)(1 1.5 e ) 888.9

4 3. (10 points) The number of claims in a calendar year under a cancer policy has the following distribution: Number of Claims Probability The amount of each claim is distributed as an exponential distribution with a mean of 100. Let S be the random variable representing the aggregate claims per insured in a calendar year. Calculate Var[ S ]. Var( S) E[ N] Var( X ) Var( N)( E[ X ]) EN [ ] (0)(0.5) (1)(0.3) (0.) 0.7 Var N [ ] (0 )(0.5) (1 )(0.3) ( )(0.) (0.7) 0.61 EX [ ] 100 Var[ X ] (100) Var S [ ] (0.7)(100) (0.61)(100) 1,886, 400

5 4. (10 points) The Wu Warranty Company provides warranties on iphones and Android phones. The probability of having a claim on an iphone is 0.5. The amount of a claim on an iphone is distributed as a Gamma distribution with 100 and 3. The probability of a claim on an Android phone is 0%. The amount of a claim on an Android phone is distributed uniformly between 100 and 300. Wu provides a warranty on 500 iphones and 1000 Android phones. Let S be the random variable representing the aggregate claims. Calculate the Var[ S ]. n ( ) [ (1 ) ] 1 Var S q q q iphone (100)(3) 300 and iphone (100) (3) 30, 000 Android ( ) ( ) 1 00 and Android Var S ( ) (500) (0.5)(30,000) (0.5)(0.75)(300) (1000) (0.)( ) (0.)(0.8)(00) 19, 54,166

6 5. Amstutz Ant Farm is studying the life expectancy of ants. One of the farms owners, Kevin, collects the following data on 100 ants: a. There were 70 ants in the farm at time 0. b. There were 15 ants that entered the farm at time c. There were 10 ants that entered the farm at time 3 d. There were 5 ants that entered the farm at time 4 e. There were 18 ants that escaped from the farm at time f. There were 1 ants that escaped from the farm at time 4 g. There were 5 ants still alive at the end of 10 days. h. The remaining 45 ants died as follows: Number of Days till Death Number of Ants Dying a. (10 points) Jana, the farm s actuary, uses the Nelson-Åalen estimator to calculate ˆ(4) S. Determine Jana s estimate. y s H (4) r S (4) e 0.638

7 b. (10 points) Molly, who is also part owner of the farm and a statistician, decides to calculate an 80% confidence interval for S (4) using the Meier-Kaplan Product Limit Estimator and the Greenwood approximation for the variance. Determine Molly s confidence interval S(4) Var (70)(68) (68)(6) (59)(48) (58)(49) ([(4)] ( ) CI (1.8) ( , )

8 6. You are given the following data regarding claims for automobile collision coverage offered by Cao Car Assurance Company: Amount of Claim Number of Claims Total Claims 0 to to , to , to 10, ,500 10,000 or more 5,000 a. (6 points) Using the Ogive, calculate F (000). 40 F 34 (1500) and F (3000) F 40 ( ) ( ) 34 (000) 0.65 ( ) 40 ( ) 40 b. (6 points) The above is based on claims with no modifications (deductible or upper limit). Cao decides to treat the data as an empirical distribution. Further, Cao decides to apply an ordinary deductible of 500 and an upper limit of 0,000. Calculate the expected amount that Cao will pay per claim with the deductible and upper limit.

9 E[ X 0, 000] E[ X 500] First, we note that no claim exceeds 15,000 because there are two claims over 10,000 and their sum is 5,000 so the maximum claim is 15,000. Therefore, 800 1, 000,500 6,500 5, ,800 E[ X 0,000] E[ X ] (3)(500) EX [ 500] E[ X 0, 000] E[ X 500]

10 7. You are given the following sample of five claims: 50, 80, 100, 300, 500 Xi 1030 Xi 358,900 a. (4 points) Calculate the unbiased estimate of the mean of the distribution from which this sample was selected Unbiased Estimator of the Mean is X X i b. (4points) Calculate the unbiased estimate of the variance of the distribution from which this sample was selected. Unbiased Estimator of the Variance X ( N)( X ) N 1 358,900 (5)(06) 4 36,380

11 8. (10 points) You are given the following sample data: X: 5, 6, 8, 10, 10, 10, 13, 17, 0, 1 You are also given: N = 10 Sum of Xi = 10 Sum of Xi = 174 You complete Hypothesis Testing with: H0: The mean is 10. H1: The mean is not 10. Calculate the z statistic, the critical value(s) assuming a significance level of 30%, and the p value. State your conclusion with regard to the Hypothesis Testing. X S X ( N)( X ) (1) N 1 9 z / Critical Value p value = Reect the null hypothesis.

12 9. (10 points) A sample of two ( X 1 and X ) is drawn from a population of a uniform distribution between 0 and. The parameter is estimated using the following estimator: ˆ X1 X Calculate the Mean Square Error in the estimate. Your answer will be in terms of. MSE Var( ˆ ) [ bias ( )] ˆ bias ( ) E[ ˆ ] E[ X X ] E[ X ] E[ X ] ˆ ˆ Var( ) Var( X1 X ) Var( X1 ) Var( X ) MSE 0 6 6