1 - Purdue University



Chapter 2

1. Model 1 is a uniform distribution from 0 to 100. Determine the table entries for a generalized uniform distribution covering the range from a to b where a < b.

2. Let X be a discrete random variable with probability function p(x) = 2(1/3)x for x = 1, 2, 3, … What is the probability that X is odd?

3. For a distribution where x > 2, you are given:

• The hazard rate function: h(x) = z2/2x , for x > 2

• F(5) = 0.84.

Calculate z.

4. FX(t) = (t2-1)/9999 for 13

Calculate:

a. E[X]

b. Var(X)

c. e(1)

d. E[(X-1)+]

e. E[X Λ 2]

f. The Median

14. (Spreadsheet) If you roll two fair die, X is the sum of the dice. Calculate:

a. E[X]

b. Var(X)

c. e(4)

d. E[(X-4)+]

e. E[X Λ 10]

f. The Mode

g. π20

15. You are given a sample of 2, 2, 3, 5, 8. For this empirical distribution, determine:

a. The mean

b. The variance

c. The standard deviation

d. The coefficient of variation

e. The skewness

f. The kurtosis

g. The probability generating function

16. The random variable X is distributed exponentially with θ = 0.2. Calculate E[e2X].

17. * Losses follow a Pareto distribution has parameters of α = 7 and θ = 10,000. Calculate e(5,000)

18. The amount of an individual claim has a Pareto distribution with θ = 8000 and α = 9. Use the central limit theorem to approximate the probability that the sum of 500 independent claims will exceed 550,000.

Chapter 4

19. The distribution function for losses from your renter’s insurance is the following:

F(x) = 1 – 0.8[1000/(1000+x)]5 – 0.2[12000/(12000+x)]3

Calculate:

a. E[X]

b. Var(X)

c. Use the normal approximation to determine the probability that the sum of 100 independent claims will not exceed 200,000.

20. * X has a Burr distribution with parameters α = 1, γ = 2, and θ = 10000.5. Y has a Pareto distribution with parameters α = 1 and θ = 1000. Z is a mixture of X and Y with equal weights on each component. Determine the median of Z.

21. The random variable X is distributed as a Pareto distribution with parameters α and θ. E(X) = 1 and Var(X) = 3. The random variable Y is equal to 2X. Calculate the Var(Y).

22. * Claim severities are modeled using a continuous distribution and inflation impacts claims uniformly at an annual rate of s. Which of the following are true statements regarding the distribution of claim severities after the effect of inflation?

i. An exponential distribution will have a scale parameter of (1+s)θ.

ii. A Pareto distribution will have scale parameters (1+s)α and (1+s)θ.

iii. An Inverse Gaussian distribution will have a scale parameter of (1+s)θ.

23. * The aggregate losses of Eiffel Auto Insurance are denoted in euro currency and follow a Lognormal distribution with μ = 8 and σ = 2. Given that 1 euro = 1.3 dollars, determine the lognormal parameters for the distribution of Eiffel’s losses in dollars.

24. (Spreadsheet) Calculate Γ(1.5).

25. The random variable X is the number of dental claims in a year and is distributed as a gamma distribution given parameter θ and with parameter α = 1. θ is distributed uniformly between 1 and 3. Calculate E(X) and Var(X).

26. A dental insurance company has 1000 insureds. Assume the number of claims from each insured is independent. Using the information in Problem 25 and the normal approximation, calculate the probability that the Company will incur more than 2100 claims.

27. * Let N have a Poisson distribution with mean Λ. Let Λ have a uniform distribution on the interval (0,5). Determine the unconditional probability that

N > 2.

28. The number of hospitalization claims in a year follows a Poisson distribution with a mean of λ. The probability of exactly three claims during a year is 60% of the probability that there will be 2 claims. Determine the probability that there will be 5 claims.

29. If the number of claims is distributed as a Poison distribution with λ = 3, calculate:

a. Pr(N = 0)

b. Pr(N = 1)

c. Pr(N = 2)

d. E(N)

e. Var(N)

30. If the number of claims is distributed as a zero truncated Poison distribution with λ = 3, calculate:

a. Pr(N = 0)

b. Pr(N = 1)

c. Pr(N = 2)

d. E(N)

e. Var(N)

31. If the number of claims is distributed as a zero modified Poison distribution with λ = 3 and p0M = 0.5, calculate:

a. Pr(N = 0)

b. Pr(N = 1)

c. Pr(N = 2)

d. E(N)

e. Var(N)

32. If the number of claims is distributed as a Geometric distribution with β = 3, calculate:

a. Pr(N = 0)

b. Pr(N = 1)

c. Pr(N = 2)

d. E(N)

e. Var(N)

33. (Spreadsheet) If the number of claims is distributed as a Binomial distribution with m = 6 and q = 0.4, calculate:

a. Pr(N = 0)

b. Pr(N = 1)

c. Pr(N = 2)

d. E(N)

e. Var(N)

34. (Spreadsheet) If the number of claims is distributed as a Negative Binomial distribution with γ = 3 and β = 2, calculate:

a. Pr(N = 0)

b. Pr(N = 1)

c. Pr(N = 2)

d. E(N)

e. Var(N)

35. (Spreadsheet) If the number of claims is distributed as a Logarithmic distribution with β = 3, calculate:

a. Pr(N = 0)

b. Pr(N = 1)

c. Pr(N = 2)

d. E(N)

e. Var(N)

36. * The Independent Insurance Company insures 25 risks, each with a 4% probability of loss. The probabilities of loss are independent. What is the probability of 4 or more losses in the same year? (Hint: Use the binomial distribution.)

37. * You are given a negative binomial distribution with γ = 2.5 and β = 5. For what value of k does pk take on its largest value?

38. * N is a discrete random variable form the (a, b, 0) class of distributions. The following information is known about the distribution:

• P(N = 0) = 0.327680

• P(N = 1) = 0.327680

• P(N = 2) = 0.196608

• E[N] = 1.25

Based on this information, which of the following are true statements?

I. P(N = 3) = 0.107965

II. N is from a binomial distribution.

III. N is from a Negative Binomial Distribution.

39. * You are given:

• Claims are reported at a Poisson rate of 5 per year.

• The probability that a claim will settle for less than 100,000 is 0.9.

What is the probability that no claim of 100,000 or more will be reported in the next three years?

40. N is distributed as a zero modified geometric distribution with β = 3 and p0M = 0.5. Calculate:

a. e(2)

b. Var[N ^ 3]

c. E[(N-2)+]

Chapter 5

41. Losses are distributed Exponentially with θ = 1000. Losses are subject to an ordinary deductible of 500. Calculate:

a. fYL(y)

b. FYL(y)

c. SYL(y)

d. E(YL)

e. The loss elimination ratio

42. Losses are distributed Exponentially with parameter θ = 1000. Losses are subject to an ordinary deductible of 500. Calculate:

a. fYP(y)

b. FYP(y)

c. E(YP)

43. Losses are distributed Exponentially with θ = 1000. Losses are subject to a franchise deductible of 500. Calculate:

a. fYL(y)

b. FYL(y)

c. SYL(y)

d. E(YL)

e. The loss elimination ratio

44. Losses are distributed Exponentially with parameter θ = 1000. Losses are subject to a franchise deductible of 500. Calculate:

a. fYP(y)

b. FYP(y)

c. E(YP)

45. Last year, losses were distributed Exponentially with θ = 1000. This year losses are subject to 10% inflation. Losses in both years are subject to an ordinary deductible of 500. Calculate the following for this year:

a. E(YL)

b. E(YP)

46. * Losses follow an exponential distribution with parameter θ. For a deductible of 100, the expected payment per loss is 2000. What is the expected payment per loss in terms of θ for a deductible of 500?

47. * In year 2007, claim amounts have the following Pareto distribution

F(x) = 1 – (800/[x+800])3

The annual inflation rate is 8%. A franchise deductible of 300 will be implemented in 2008. Calculate the loss elimination ration of the franchise deductible.

48. (Spreadsheet) If you roll two fair die, X is the sum of the dice. Let X represent the distribution of losses from a particular event. If an ordinary deductible of 3 is applied, calculate:

a. E(YL)

b. E(YP)

c. Var(YL)

d. Var(YP)

49. Losses are distributed uniformly between 0 and 100,000. An insurance policy which covers the losses has a 10,000 deductible and an 80,000 upper limit. The upper limit is applied prior to applying the deductible. Calculate:

a. E(YL)

b. E(YP)

c. Var(YL)

d. Var(YP)

50. Last year, losses were distributed Exponentially with θ = 1000. This year losses are subject to 25% inflation. Losses in both years are subject to an ordinary deductible of 500, an upper limit of 4000, and coinsurance of 20% Calculate the following for this year:

a. E(YL)

b. E(YP)

51. * An insurance company offers two types of policies: Type Q and Type R. Type Q has no deductible, but has a policy limit of 3000. Type R has no limit, but has an ordinary deductible of d. Losses follow a Pareto distribution with θ = 2000 and α = 3.

Calculate the deductible d such that both policies have the same expected cost per loss.

52. * Well Traveled Insurance Company sells a travel insurance policy that reimburses travelers for any expense incurred for a planned vacation that is cancelled because of airline bankruptcies. Individual claims follow a Pareto distribution with α = 2 and θ = 500. Well Traveled imposes a limit of 1000 on each claim. If a planned policyholder’s planned vacation is cancelled due to airline bankruptcy and he or she has incurred more than 1000 of expenses, what is the expected non-reimbursable amount of the claim?

53. If X is uniformly distributed on from 0 to b. An ordinary deductible of d is applied. Calculate:

a. E[YL]

b. Var[YL]

c. E[YP]

d. Var[YP]

54. Losses follow a Pareto distribution with α = 3 and θ = 200. A policy covers the loss except for a franchise deductible of 50. X is the random variable representing the amount paid by the insurance company per payment. Calculate the expected value of the X.

55. Losses in 2007 are uniformly distributed between 0 and 100,000. An insurance policy pays for all claims in excess of an ordinary deductible of $20,000. For 2008, claims will be subject to uniform inflation of 20%. The Company implements an upper limit u (without changing the deductible) so that the expected cost per loss in 2008 is the same as the expected cost per loss in 2007. Calculate u.

56. Losses follow an exponential distribution with a standard deviation of 100. An insurance company applies a policy deductible d which results in a Loss Elimination Ratio of 0.1813. Calculate d.

57. Losses are distributed following the single parameter Pareto (Not the same as the Pareto distribution in Question 54) with α = 4 and θ = 90. An insurance policy is issued with a deductible of 100. Calculate the Var(YL).

58. Losses follow a Pareto distribution with α = 5 and θ = 2000. A policy pays 100% of losses from 500 to 1000. In other words, if a loss occurs for less than 500, no payment is made. If a loss occurs which exceeds 1000, no payment is made. However, if a loss occurs which is between 500 and 1000, then the entire amount of the loss is paid. Calculate E(YP).

59. * You are given the following:

• Losses follow a lognormal distribution with parameters μ =10 and σ = 1.

• For each loss less than or equal to 50,000, the insurer makes no payment.

• For each loss greater than 50,000, the insurer pays the entire amount of the loss up to the maximum covered loss of 100,000.

Determine the expected amount of the loss.

60. You are given the following:

• Losses follow a distribution prior to the application of any deductible with a mean of 2000.

• The loss elimination ratio at a deductible of 1000 is 0.3.

• 60% of the losses in number are less than the deductible of 1000.

Determine the average size of the loss that is less than the deductible of 1000.

61. Losses follow a Pareto distribution with α = 5 and θ = 2000. An insurance policy covering these losses has a deductible of 100 and makes payments directly to the physician. Additionally, the physician is entitled to a bonus if the loss is less than 500. The bonus is 10% of the difference between 500 and the amount of the loss.

The following table should help clarify the arrangement:

|Amount of Loss |Loss Payment |Bonus |

|50 |0 |45 |

|100 |0 |40 |

|250 |150 |25 |

|400 |300 |10 |

|500 |400 |0 |

|1000 |900 |0 |

Calculate the expected total payment (loss payment plus bonus) from the insurance policy to the physician per loss.

Chapter 6

62. * For an insured, Y is the random variable representing the total amount of time spent in a hospital each year. The distribution of the number of hospital admissions in a year is:

|Number of Admissions |Probability |

|0 |0.60 |

|1 |0.30 |

|2 |0.10 |

The distribution of the length of each stay for an admission follows a Gamma distribution with α = 1 and θ = 5.

Calculate E[Y] and Var[Y].

63. For an insurance company, each loss has a mean of 100 and a variance of 100. The number of losses follows a Poisson distribution with a mean of 500. Each loss and the number of losses are mutually independent.

The loss ratio for the insurance company is defined as the ratio of aggregate losses to the total premium collected.

The premium collected is 110% of the expected aggregate losses.

Using the normal approximation, calculate the probability that the loss ratio will exceed 95%.

64. An automobile insurer has 1000 cars covered during 2007. The number of automobile claims for each car follows a negative binomial distribution with β = 1 and γ = 1.5. Each claim is distributed exponentially with a mean of 5000. Assume that the number of claims and the amount of the loss are independent and identically distributed.

Using the normal distribution as an approximating distribution of aggregate losses, calculate the probability that losses will exceed 8 million.

65. X1, X2, and X3 are mutually independent random variables. These random variables have the following probability functions:

|x |f1(x) |f2(x) |f3(x) |

|0 |0.6 |0.4 |0.1 |

|1 |0.4 |0.3 |0.4 |

|2 |0.0 |0.2 |0.5 |

|3 |0.0 |0.1 |0.0 |

Calculate fS(x).

66. The number of claims follows a Poisson distribution with a mean of 3. The distribution of claims is fX(1) = 1/3 and fX(2) = 2/3. Calculate fS(4).

67. (Spreadsheet) The number of claims and the amount of each claim is mutually independent. The frequency distribution is:

|n |pn |

|1 |0.05 |

|2 |0.20 |

|3 |0.30 |

|4 |0.25 |

|5 |0.15 |

|6 |0.05 |

The severity distribution is:

|x |fX(x) |

|100 |0.25 |

|200 |0.20 |

|300 |0.15 |

|400 |0.12 |

|500 |0.10 |

|600 |0.08 |

|700 |0.06 |

|800 |0.04 |

Calculate fS(x). (See Example 6.6 in the book for an example.)

68. You are given the following table for aggregate claims:

|S |FS(s) |E[(X-d)+] |

|0 | | |

|100 | | |

|200 |0.50 | |

|300 |0.65 |60 |

|400 |0.75 | |

|500 | | |

|600 | | |

Losses can only occur in multiples of 100. Calculate the net stop loss premium for stop loss insurance covering losses in excess of 325.

69. * Losses follow a Poisson frequency distribution with a mean of 2 per year. The amount of a loss is 1, 2, or 3 with each having a probability of 1/3. Loss amounts are independent of the number of losses and from each other.

An insurance policy covers all losses in a year subject to an annual aggregate deductible of 2. Calculate the expected claim payments for this insurance policy.

70. Purdue University has decided to provide a new benefit to each class at the university. Each class will be provided with group life insurance. Students in each class will have 10,000 of coverage while the professor for the class will have 20,000. STAT 490C has 27 students all age 22 split 12 males and 15 females. The class also has an aging professor. The probability of death for each is listed below:

|Age |Gender |Probability of Death |

|22 |Male |.005 |

|22 |Female |.003 |

|52 |Male |.020 |

Purdue purchases a Stop Loss Policy such with an aggregate deductible of 20,000. Calculate the net premium that Purdue will pay for the stop loss coverage.

71. For an automobile policy, the severity distribution is Gamma with α = 3 and θ = 1000. The number of claims in a year is distributed as follows:

|Number of Claims |Probability |

|1 |.65 |

|2 |.20 |

|3 |.10 |

|4 |.05 |

Calculate:

a. E(S)

b. Var(S)

c. MS(t)

72. (Spreadsheet) You are given that the number of losses follow a Poisson with λ = 6. Losses are distributed as follows: fX(1) = fX(2) = fX(4) = 1/3. Find fS(x) for total losses of 100 or less.

73. On a given day, a doctor provides medical care for NA adults and NC children. Assume that NA and NC have Poisson distributions with parameters 3 and 2 respectively. The distributions for the length of care per patient are as follows:

|Time |Adult |Child |

|1 hour |0.4 |0.9 |

|2 hours |0.6 |0.1 |

Let NA, NC, and the lengths of stay for all patients be independent. The doctor charges 200 per hour. Determine the probability that the office income on a given day will be less than or equal to 800.

74. (Spreadsheet) Losses follow a Pareto distribution with α = 1 and θ = 1000.

a. Using the Method of Rounding with a span of 100, calculate fj for 0 < j 17.

Calculate the z statistic, the critical value(s) assuming a significance level of 1%, and the p value. State your conclusion with regard to the Hypothesis Testing.

Wang Warranty Corporation is testing iPods. Wang starts with 100 iPods and tests them by dropping them on the ground. Wang records the number of drops before each iPod will no longer play. The following data is collected from this test:

|Drops to Failure |Number |Drops to Failure |Number |

|1 |6 |9 |7 |

|2 |2 |10 |6 |

|4 |3 |11 |7 |

|6 |4 |12 |7 |

|7 |4 |13 |48 |

|8 |5 |22 |1 |

Ledbetter Life Insurance Company is completing a mortality study on a 3 year term insurance policy. The following data is available:

|Life |Date of Entry |Date of Exit |Reason for Exit |

|1 |0 |0.2 |Lapse |

|2 |0 |0.3 |Death |

|3 |0 |0.4 |Lapse |

|4 |0 |0.5 |Death |

|5 |0 |0.5 |Death |

|6 |0 |0.5 |Lapse |

|7 |0 |1.0 |Death |

|8 |0 |3.0 |Expiry of Policy |

|9 |0 |3.0 |Expiry of Policy |

|10 |0 |3.0 |Expiry of Policy |

|11 |0 |3.0 |Expiry of Policy |

|12 |0 |3.0 |Expiry of Policy |

|13 |0 |3.0 |Expiry of Policy |

|14 |0 |3.0 |Expiry of Policy |

|15 |0 |3.0 |Expiry of Policy |

|16 |0.5 |2.0 |Lapse |

|17 |0.5 |1.0 |Death |

|18 |1.0 |3.0 |Expiry of Policy |

|19 |1.0 |3.0 |Expiry of Policy |

|20 |2.0 |2.5 |Death |

Schneider Trucking Company had the following losses during 2006:

|Amount |Number |Total Amount |

|of Claim |of Payments |of Losses |

|0 – 10 |8 |60 |

|10-20 |5 |70 |

|20-30 |4 |110 |

|30 + |3 |200 |

| | | |

|Total |20 |440 |

99. Using the data from Wang Warranty Corporation, calculate:

a. p100(x)

b. F100(x)

c. The empirical mean

d. The empirical variance

e. Ĥ(x) where Ĥ(x) is the cumulative hazard function from the Nelson Åalen estimate

f. Ŝ(x) where Ŝ(x) is the survival function from the Nelson Åalen estimate

100. Using the data from Schneider Trucking Company, calculate:

a. The ogive, F20(x)

b. The histogram, f20(x)

c. E(XΛ30) minus E[(X-20)+]

101. * You are given:

|Claim Size (X) |Number of Claims |

|(0,25] |30 |

|(25,50] |32 |

|(50,100] |20 |

|(100,200] |8 |

Assume a uniform distribution of claim sizes within each interval.

Estimate the mean of the claim size distribution.

Estimate the second raw moment of the claim size distribution.

Chapter 11

102. A mortality study is completed on 30 people. The following deaths occur during the five years:

3 deaths at time 1.0

4 deaths at time 2.0

5 deaths at time 3.0

8 deaths at time 3.8

10 deaths at time 4.5

There were no other terminations and no lives entered the study after the start of the study.

The data was smoothed using a uniform kernel with a bandwidth of 1. Calculate [pic](x) and [pic](x) for all x > 0.

103. * From a population having a distribution function F, you are given the following sample:

2.0, 3.3, 3.3, 4.0, 4.0, 4.7, 4.7, 4.7

Calculate the kernel density estimate of F(4), using a uniform kernel with bandwidth of 1.4.

104. * You study five lives to estimate the time from the onset of a disease until death. The times to death are:

2 3 3 3 7

Using a triangular kernel with a bandwidth of 2, estimate the density function at 2.5.

105. You are given the following random sample:

12 15 27 42

The data is smoothed using a uniform kernel with a bandwidth of 6.

Calculate the mean and variance of the smoothed distribution.

106. You are given the following random sample:

12 15 27 42

The data is smoothed using a triangular kernel with a bandwidth of 12.

Calculate the mean and variance of the smoothed distribution.

107. You are given the following random sample:

12 15 27 42

The data is smoothed using a gamma kernel with a bandwidth of 3.

Calculate the mean and variance of the smoothed distribution.

Chapter 12

108. You are given the following sample of claims obtained from an inverse gamma distribution:

X: 12, 13, 16, 16, 22, 24, 26, 26, 28, 30

The sum of X is 213 and the sum of X2 is 4921.

Calculate α and θ using the method of moments.

109. * You are given the following sample of five claims:

4 5 21 99 421

Find the parameters of a Pareto distribution using the method of moments.

110. * A random sample of death records yields the follow exact ages at death:

30 50 60 60 70 90

The age at death from which the sample is drawn follows a gamma distribution. The parameters are estimated using the method of moments.

Determine the estimate of α.

111. * You are given the following:

• The random variable X has the density function

f(x) = αx-α-1 , 1 < x < ∞, α >1

• A random sample is taken of the random variable X.

Calculate the estimate of α in terms of the sample mean using the method of moments.

112. * You are given the following:

• The random variable X has the density function f(x) = {2(θ – x)}/θ2, 0 < x < θ

• A random sample of two observations of X yields values of 0.50 and 0.70.

Determine θ using the method of moments.

113. You are given the following sample of claims:

X: 12, 13, 16, 16, 22, 24, 26, 26, 28, 30

Calculate the smoothed empirical estimate of the 40th percentile of this distribution.

114. * For a complete study of five lives, you are given:

a. Deaths occur at times t = 2, 3, 3, 5, 7.

b. The underlying survival distribution S(t) = 4-λt, t > 0

Using percentile matching at the median, calculate the estimate of λ.

115. You are given the following 9 claims:

X: 10, 60, 80, 120, 150, 170, 190, 230, 250

The sum of X = 1260 and the sum of X2 = 227,400.

The data is modeled using an exponential distribution with parameters estimated using the percentile matching method.

Calculate θ based on the empirical value of 120.

116. * For a sample of 10 claims, x1 < x2 < … < x10 you are given:

a. The smoothed empirical estimate of the 55th percentile is 380.

b. The smoothed empirical estimate of the 60th percentile is 402.

Determine x6.

117. You are given the following:

a. Losses follow a Pareto distribution with parameters α and θ.

b. The 10th percentile of the distribution is θ – k, where k is a constant.

c. The 90th percentile of the distribution is 5θ – 3k.

Determine α.

118. You are given the following random sample of 3 data points from a population with a Pareto distribution with θ = 70:

X: 15 27 43

Calculate the maximum likelihood estimate for α.

119. * You are given:

a. Losses follow an exponential distribution with mean θ.

b. A random sample of 20 losses is distributed as follows:

|Range |Frequency |

|[0,1000] |7 |

|(1000, 2000) |6 |

|(2000,∞) |7 |

Calculate the maximum likelihood estimate of θ.

120. * You are given the following:

• The random variable X has the density function f(x) = {2(θ – x)}/θ2, 0 < x < θ

• A random sample of two observations of X yields values of 0.50 and 0.90.

Determine the maximum likelihood estimate for θ.

121. * You are given:

a. Ten lives are subject to the survival function S(t) = (1-t/k)0.5, 0 < t < k

b. The first two deaths in the sample occured at time t = 10.

c. The study ends at time t = 10.

Calculate the maximum likelihood estimate of k.

122. * You are given the following:

a. The random variable X follows the exponential distribution with parameter θ.

b. A random sample of three observations of X yields values of 0.30, 0.55, and 0.80

Determine the maximum likelihood estimate of θ.

123. * Ten laboratory mice are observed for a period of five days. Seven mice die during the observation period, with the following distribution of deaths:

|Time of Death in Days |Number of Deaths |

|2 |1 |

|3 |2 |

|4 |1 |

|5 |3 |

The lives in the study are subject to an exponential survival function with mean of θ.

Calculate the maximum likelihood estimate of θ.

124. * A policy has an ordinary deductible of 100 and a policy limit of 1000. You observe the following 10 payments:

15 50 170 216 400 620 750 900 900 900

An exponential distribution is fitted to the ground up distribution function, using the maximum likelihood estimate.

Determine the estimated parameter θ.

125. * Four lives are observed from time t = 0 until death. Deaths occur at t = 1, 2, 3, and 4. The lives are assumed to follow a Weibull distribution with τ = 2.

Determine the maximum likelihood estimator for θ.

126. * The random variable X has a uniform distribution on the interval [0,θ]. A random sample of three observations of X are recorded and grouped as follows:

| |Number of Observations |

|Interval | |

|[0,k) |1 |

|[k,5) |1 |

|[5,θ] |1 |

Calculate the maximum likelihood estimate of θ

127. * A random sample of three claims from a dental insurance plan is given below:

225 525 950

Claims are assumed to follow a Pareto distribution with parameters θ = 150 and α.

Determine the maximum likelihood estimate of α.

128. * The following claim sizes are experienced on an insurance coverages:

100 500 1,000 5,000 10,000

You fit a lognormal distribution to this experience using maximum likelihood.

Determine the resulting estimate of σ.

You are given the following data from a sample:

|k |nk |

|0 |20 |

|1 |25 |

|2 |30 |

|3 |15 |

|4 |8 |

|5 |2 |

Use this data for the next four problems.

129. Assuming a Binomial Distribution, estimate m and q using the Method of Moments.

130. Assuming a Binomial Distribution, find the MLE of q given that m = 6.

131. (Spreadsheet) Assuming a Binomial Distribution, find the MLE of m and q.

132. Assuming a Poisson Distribution, approximate the 90% confidence interval for the true value of λ.

133. You are given the following 20 claims:

X: 10, 40, 60, 65, 75, 80, 120, 150, 170, 190, 230, 340, 430, 440, 980, 600, 675, 950, 1250, 1700

The data is being modeled using an exponential distribution with θ = 427.5.

Calculate D(200).

134. You are given the following 20 claims:

X: 10, 40, 60, 65, 75, 80, 120, 150, 170, 190, 230, 340, 430, 440, 980, 600, 675, 950, 1250, 1700

The data is being modeled using an exponential distribution with θ = 427.5.

You are developing a p-p plot for this data. What are the coordinates for x7 = 120.

135. Mark the following statements True or False with regard to the Kolmogorov-Smirnov test:

The Kolmogorov-Smirnov test may be used on grouped data as well as individual data.

If the parameters of the distribution being tested are estimated, the critical values do not need to be adjusted.

If the upper limit is less than ∞, the critical values need to be larger.

136. Balog’s Bakery has workers’ compensation claims during a month of:

100, 350, 550, 1000

Balog’s owner, a retired actuary, believes that the claims are distributed exponentially with θ = 500.

He decides to test his hypothesis at a 10% significance level.

Calculate the Kolmogorov-Smirnov test statistic.

State the critical value for his test and state his conclusion.

He also tests his hypothesis using the Anderson-Darling test statistic. State the values of this test statistic under which Mr. Schmidt would reject his hypothesis.

137. * The observations of 1.7, 1.6, 1.6, and 1.9 are taken from a random sample. You wish to test the goodness of fit of a distribution with probability density function given by f(x) = 0.5x for 0 < x < 2.

Using the Kolmogorov-Smirnov statistic, which of the following should you do?

a. Accept at both levels

b. Accept at the 0.01 level but reject at the 0.10 level

c. Accept at the 0.10 level but reject at the 0.01 level

d. Reject at both levels

e. Cannot be determined.

138. * Two lives are observed beginning at time t=0. One dies at time 5 and the other dies at time 9. The survival function S(t) = 1 – (t/10) is hypothesized.

Calculate the Kolmogorov-Smirnov statistic.

139. * From a laboratory study of nine lives, you are given:

a. The times of death are 1, 2, 4, 5, 5, 7, 8, 9, 9

b. It has been hypothesized that the underlying distribution is uniform

with ω = 11.

Calculate the Kolmogorov-Smirnov statistic for the hypothesis.

140. You are given the following data:

|Claim Range |Count |

|0-100 |30 |

|100-200 |25 |

|200-500 |20 |

|500-1000 |15 |

|1000+ |10 |

H0: The data is from a Pareto distribution.

H1: The data is not from a Pareto distribution.

Your boss has used the data to estimate the parameters as α = 4 and θ = 1200.

Calculate the chi-square test statistic.

Calculate the critical value at a 10% significance level.

State whether you would reject the Pareto at a 10% significance level.

141. During a one-year period, the number of accidents per day in the parking lot of the Steenman Steel Factory is distributed:

|Number of Accidents |Days |

|0 |220 |

|1 |100 |

|2 |30 |

|3 |10 |

|4+ |5 |

H0: The distribution of the number of accidents is distributed as Poison with a mean of 0.625.

H1: The distribution of the number of accidents is not distributed as Poison with a mean of 0.625.

Calculate the chi-square statistic.

Calculate the critical value at a 10% significance level.

State whether you would reject the H0 at a 10% significance level.

142. * You are given the following random sample of automobile claims:

54 140 230 560 600 1,100 1,500 1,800 1,920 2,000

2,450 2,500 2,580 2,910 3,800 3,800 3,810 3,870 4,000 4,800

7,200 7,390 11,750 12,000 15,000 25,000 30,000 32,200 35,000 55,000

You test the hypothesis that automobile claims follow a continuous distribution F(x) with the following percentiles:

|x |310 |500 |2,498 |4,876 |7,498 |12,930 |

|F(x) |0.16 |0.27 |0.55 |0.81 |0.90 |0.95 |

You group the data using the largest number of groups such that the expected number of claims in each group is at least 5.

Calculate the Chi-Square goodness-of-fit statistic.

143. Based on a random sample, you are testing the following hypothesis:

H0: The data is from a population distributed binomial with m = 6 and q = 0.3.

H1: The data is from a population distributed binomial.

You are also given:

L(θ0) = .1 and L(θ1) = .3

Calculate the test statistic for the Likelihood Ratio Test

State the critical value at the 10% significance level

144. State whether the following are true or false

• The principle of parsimony states that a more complex model is better because it will always match the data better.

• In judgment-based approaches to determining a model, a modeler’s experience is critical.

• In most cases, judgment is required in using a score-based approach to selecting a model.

Chapter 17

145. A random number generated from a uniform distribution on (0, 1) is 0.6. Using the inverse transformation method, calculate the simulated value of X assuming:

• X is distributed Pareto with α = 3 and θ = 2000

0.5x 0 < x < 1.2

• F(x) = 0.6 1.2 < x < 2.4

0.5x -0.6 2.4 < x < 3.2

• F(x) = 0.1x -1 10 < x < 15

0.05x 15 < x < 20

146. * You are given that f(x) = (1/9)x2 for 0 < x < 3.

You are to simulate three observations from the distribution using the inversion method. The follow three random numbers were generated from the uniform distribution on [0,1]:

0.008 0.729 0.125

Using the three simulated observations, estimate the mean of the distribution.

147. * You are to simulate four observations from a binomial distribution with two trials and probability of success of 0.30. The following random numbers are generated from the uniform distribution on [0,1]:

0.91 0.21 0.72 0.48

Determine the number of simulated observations for which the number of successes equals zero.

148. Kyle has an automobile insurance policy. The policy has a deductible of 500 for each claim. Kyle is responsible for payment of the deductible.

The number of claims follows a Poison distribution with a mean of 2. Automobile claims are distributed exponentially with a mean of 1000.

Kyle uses simulation to estimate the claims. A random number is first used to calculate the number of claims. Then each claim is estimated using random numbers using the inverse transformation method.

The random numbers generated from a uniform distribution on (0, 1) are 0.7, 0.1, 0.5, 0.8, 0.3, 0.7, 0.2.

Calculate the simulated amount that Kyle would have to pay in the first year.

149. * Insurance for a city’s snow removal costs covers four winter months.

You are given:

• There is a deductible of 10,000 per month.

• The insurer assumes that the city’s monthly costs are independent and normally distributed with mean of 15,000 and standard deviation of 2000.

• To simulate four months of claim costs, the insurer uses the inversion method (where small random numbers correspond to low costs).

• The four numbers drawn from the uniform distribution on [0,1] are:

0.5398 0.1151 0.0013 0.7881

Calculate the insurer’s simulated claim cost.

150. * Annual dental claims are modeled as a compound Poisson process where the number of claims has mean of 2 and the loss amounts have a two-parameter Pareto distribution with θ = 500 and α = 2.

An insurance pays 80% of the first 750 and 100% of annual losses in excess of 750.

You simulate the number of claims and loss amounts using the inversion method.

The random number to simulate the number of claims is 0.80. The random numbers to simulate the amount of claims are 0.60, 0.25, 0.70, 0.10, and 0.80.

Calculate the simulated insurance claims for one year.

151. A sample of two selected from a uniform distribution over (1,U) produces the following values:

3 7

You estimate U as the Max(X1, X2).

Estimate the Mean Square Error of your estimate of U using the bootstrap method.

152. * Three observed values from the random variable X are:

1 1 4

You estimate the third central moment of X using the estimator:

g(X1, X2, X3) = 1/3 Σ(Xj -[pic])3

Determine the bootstrap estimate of the mean-squared error of g.

Answers

1. Not Provided

2. ¾

3. 2

4. 100/9999

5.

a. 0.725

b. 0.093

6.

a. 0.822

b. 0.244

7. No Answer Given

8. 1/√5

9.

a. (bk+1 – ak+1)/(b-a)(k+1)

b. (b+a)/2

c. (b-a)2/12

d. (b-d)/2

10.

a. θ/(α-1)

b. αθ2/[(α-1)2(α-2)]

c. [α/( α-2)]0.5

11.

a. θα

b. θ2α

c. 1/√α

d. 2/√α

12.

a. θ

b. θ2

c. θ

13.

a. 2.25

b. 27/80

c. 17/13

d. 34/27

e. 50/27

f. 2.3811

Answers

14.

a. 7

b. 5.8333

c. 3.7333

d. 3.1111

e. 6.8888

f. 7

g. 5

15.

a. 4

b. 5.2

c. 2.28

d. 0.57

e. 0.8096

f. 2.145

g. 0.4z2 + 0.2z3 + 0.2z5 + 0.2z8

16. 5/3

17. 2500

18. 0.0244

19.

a. 1400

b. 26,973,333

c. 87.7%

20. 100

21. 12

22. i only. Be sure to indicate how to make the others true.

23. μ = 8.26 and σ = 2.00

24. 0.886227

25. 2 and 14/3

26. 7.2%

27. 0.6094

28. 2.6%

29.

a. 0.0498

b. 0.1494

c. 0.2240

d. 3

e. 3

30.

a. 0

b. 0.1572

c. 0.2358

d. 3.1572

e. 2.6609

Answers

31.

a. 0.5000

b. 0.0786

c. 0.1179

d. 1.5786

e. 3.8224

32.

a. 0.2500

b. 0.1875

c. 0.1406

d. 3

e. 12

33.

a. 0.0467

b. 0.1866

c. 0.3110

d. 2.4

e. 1.44

34.

a. 0.0370

b. 0.0741

c. 0.0988

d. 6

e. 18

35.

a. 0

b. 0.5410

c. 0.2029

d. 2.1640

e. 3.9731

36. 0.0165

37. 7

38. Statement III is only true statement

39. 0.2231

40.

a. 4

b. 1.6943

c. 1.125

Answers

41.

a. fYL(y) = 1 – e-0.5, y = 0

fYL(y) = 0.001e-(0.001y + 0.5), y > 0

b. FYL(y) = 1 – e-0.5, y = 0

FYL(y) = 1 – e-(.001 y + 0.5), y > 0

c. SYL(y) = e-0.5, y = 0

SYL(y) = e-(.001 y + 0.5), y > 0

d. 1000e-0.5

e. 1 – e-0.5

42.

a. fYP(y) = 0.001e-0.001y

b. FYP(y) = 1 – e-.001 y

c. 1000

43.

a. fYL(y) = 1 – e-0.5, y = 0

fYL(y) = 0.001e-0.001y, y > 0

b. FYL(y) = 1 – e-0.5, y = 0

FYL(y) = 1 – e-.001 y, y > 0

c. SYL(y) = e-0.5, y = 0

SYL(y) = e-.001 y , y > 0

d. 1500e-0.5

e. 1 – 1.5e-0.5

44.

a. fYP(y) = 0.001e-0.001y+0.5

b. FYP(y) = 1 – e-.001 y+0.5

c. 1500

Answers

45.

a. 1100e-(0.5/1.1)

b. 1100

46. 2000e-400/θ

47. 0.165

48.

a. 4.0278

b. 4.39

c. 5.58

d. 4.48

49.

a. 38,500

b. 42,777.78

c. 641,083,333

d. 529,320,798

50.

a. 629.56

b. 939.19

51. 182.18

52. 1500

53.

a. (b-d)2/2b

b. [(b-d)3/3b] – [(b-d)2/2b]2

c. (b-d)/2

d. (b-d)2/12

54. 175

55. 71,833.62

56. 20

57. 1708.70

58. 705.06

59. 16,224

60. 333.33

61. 431.83

62. 2.5 and 23.75

63. 0.1587

64. 0.068

65.

|x |fS(x) |

| | |

|0 |0.024 |

|1 |0.130 |

|2 |0.280 |

|3 |0.280 |

|4 |0.180 |

|5 |0.086 |

|6 |0.020 |

66. 0.151436

67. None provided

68. 51.25

69. 2.360894

70. 22.60

71.

a. 4,650

b. 11,377,500

c. 0.65(1-1000t)-3 + 0.2(1-1000t)-6 + 0.1(1-1000t)-9 + 0.05(1-1000t)-12

72. No Answer Provided

73. 0.2384

74.

a. Mean = 498.61

b. Mean = 499.85

75. 0.106978 and 0.060129

76. 0.1517

77. 76

78. 165

79. 0.15625

80. 0.406

81. 0.224

82. 92.16

83. 2.25

84. 5,727

85. 46.13

86. 2.1

87. 1975

88. 0.954

89. 0.30

90. 0.543

91. 0.2775

92. 0.6254

93. (n+8)/[18(n-1)2]

94. 151.52

95. 1/5

96. -0.24

97. 5

98. z = 2.0814; critical value = 2.33; Since 2.0814 is less than 2.33, we cannot reject the null hypothesis; p = 0.0188

99.

|x |p100(x) |x |F100(x) |Ĥ(x) |Ŝ(x) |

| | |x ................
................

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