Student Solutions Manual to Accompany Loss Models: From Data to Decisions, Fourth Edition

CHAPTER 3

CHAPTER 3 SOLUTIONS

3.1   SECTION 3.1

3.1

3.2 For Model 1, σ² = 3,333.33 − 50² = 833.33, σ = 28.8675.

For Model 2, σ² = 4,000,000 − 1,000² = 3,000,000, σ = 1,732.05. μ′₃ and μ′₄ are both infinite so the skewness and kurtosis are not defined.

For Model 3, σ² = 2.25 − .93² = 1.3851, σ = 1.1769.

For Model 4, σ² = 6,000,000,000 − 30,000² = 5,100,000,000, σ = 71,414.

For Model 5, σ² = 2,395.83 − 43.75² = 481.77, σ = 21.95.

3.3 The standard deviation is the mean times the coefficient of variation, or 4, and so the variance is 16. From (3.3) the second raw moment is 16 + 2² = 20. The third central moment is (using Exercise 3.1) 136 − 3(20)(2) + 2(2)³ = 32. The skewness is the third central moment divided by the cube of the standard deviation, or 32/4³ = 1/2.

3.4 For a gamma distribution the mean is αθ. The second raw moment is α(α + 1)θ², and so the variance is αθ². The ...