1 - Purdue University



Chapter 2

1. Model 1 in the table handed out in class is a uniform distribution from 0 to 100. Determine what the table entries would be 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

g. The standard deviation principle with k = 1

h. VaR.80

i. TVaR.80

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. VaR.80

h. TVaR.80

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

17. 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.

18. Lifetimes of an iPod follows a Single Parameter Pareto distribution with [pic] and [pic] The expected lifetime of an iPod is 8 years.

Calculate the probability that the lifetime of an iPod is at least 6 years.

19. You are given that [pic] .

You are also given that [pic] is:

|[pic] |[pic] |

|100 |0.4 |

|200 |0.3 |

|300 |0.2 |

|400 |0.1 |

Calculate [pic] .

20. A company has 50 employees whose dental expenses are mutual independent. For each employee, the company reimburses 100% of the dental expenses. The dental expense for each employee is distributed as follows:

|Expense |Probability |

|0 |0.5 |

|100 |0.3 |

|400 |0.1 |

|900 |0.1 |

Using the normal approximation, calculate the 95th percentile of the cost to the company.

Chapter 4

21. 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.

22. * 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.

23. 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).

24. * 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.

25. * 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.

Chapter 5

26. (Spreadsheet) Calculate Γ(1.2).

27. 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).

28. A dental insurance company has 1000 insureds. Assume the number of claims from each insured is independent. Using the information in Problem 27 and the normal approximation, calculate the probability that the Company will incur more than 2100 claims.

29. * 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.

Chapter 6

30. 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.

31. 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)

32. 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)

33. 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)

34. 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)

35. 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)

36. 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)

37. * 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.)

38. * 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?

39. * N is a discrete random variable from 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.

40. * 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?

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

a. e(2)

b. [pic]

c. E[(N-2)+]

42. N is the random variable representing the number of claims under homeowners insurance. N is distributed as a zero modified geometric distribution with β = 2. [pic] = 2/27. Calculate [pic] .

43. Under an unmodified geometric distribution, [pic]

Under a zero-modified geometric distribution, [pic]

The parameter [pic] is the same for both distributions.

Calculate [pic] .

Answers

1. Not Provided

2. ¾

3. 2

4. 100/9999

5.

a. 0.725

b. 0.093

6.

a. .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

e. (1-p)a+pb

f. [b+(1-p)a+pb]/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

g. 2.83

h. 2.785

i. 2.895

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. 5

h. 8

16. 2500

17. 0.0244

18. 4/9

19. 7777.78

20. 11,173

21.

a. 1400

b. 26,973,333

c. 87.7%

22. 100

23. 12

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

25. μ = 8.26 and σ = 2.00

26. 0.918169

27. 2 and 14/3

28. 7.2%

29. 0.6094

30. 2.6%

31.

a. 0.0498

b. 0.1494

c. 0.2240

d. 3

e. 3

32.

a. 0

b. 0.1572

c. 0.2358

d. 3.1572

e. 2.6609

Answers

33.

a. 0.5000

b. 0.0786

c. 0.1179

d. 1.5786

e. 3.8224

34.

a. 0.2500

b. 0.1875

c. 0.1406

d. 3

e. 12

35.

a. 0.0467

b. 0.1866

c. 0.3110

d. 2.4

e. 1.44

36.

a. 0.0370

b. 0.0741

c. 0.0988

d. 6

e. 18

37. 0.0165

38. 7

39. Statement III is only true statement

40. 0.2231

41.

a. 4

b. 1.6943

c. 1.125

42. 1/6

43. 0.1

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download