### Practice Problem Set 1 – working with moments

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This post has two practice problems to complement the previous post Working with moments.

 Practice Problem 1a Losses are modeled by a distribution that is the independent sum of two exponential distributions, one with mean 6 and the other with mean 12. Calculate the skewness of the loss distribution. Calculate the kurtosis of the loss distribution.
 Practice Problem 1b Losses are modeled by a distribution that is a mixture of two exponential distributions, one with mean 6 and the other with mean 12. The weight of each distribution is 50%. Calculate the skewness of the loss distribution. Calculate the kurtosis of the loss distribution.

Comment
As the previous post Working with moments shows, this is primarily an exercise in finding moments. There is no need to first find the probability density function of the loss distribution. For more information on exponential distribution, see here.

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Solutions

Problem 1a

$E(X)=18$

$E(X^2)=504$

$E(X^3)=19440$

$E(X^4)=964224$

$\displaystyle \gamma_1=\frac{3888}{180^{1.5}}=1.61$

$\displaystyle \text{Kurt}[X]=\frac{229392}{180^{2}}=7.08$

Problem 1b

$E(X)=9$

$E(X^2)=180$

$E(X^3)=5832$

$E(X^4)=264384$

$\displaystyle \gamma_1=\frac{2430}{99^{1.5}}=2.4669$

$\displaystyle \text{Kurt}[X]=\frac{122229}{99^{2}}=12.47$

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$\copyright$ 2017 – Dan Ma