**3.1**

**3.2** For Model 1, *σ*^{2} = 3,333.33 − 50^{2} = 833.33, *σ* = 28.8675.

For Model 2, *σ*^{2} = 4,000,000 − 1,000^{2} = 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, *σ** ^{2}* = 2.25 − .93

For Model 4, *σ** ^{2}* = 6,000,000,000 − 30,000

For Model 5, *σ** ^{2}* = 2,395.83 − 43.75

**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 + 2^{2} = 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 32/4^{3} = 1/2.

**3.4** For a gamma distribution the mean is *α**θ*. The second raw moment is *α*(*α* + 1)*θ*^{2}, and so the variance is *αθ*^{2}. The ...

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