Practice Problem Set 1 – working with moments
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.

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%.

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.
Solutions
Problem 1a
Problem 1b
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2017 – Dan Ma
Working with moments
This post gives some background information on calculation involving moments.
Let be a random variable. Let be its mean and be its variance. Thus is the standard deviation of .
The expectation is the th raw moment. It is also called the th moment about zero. The expectation is the th central moment. It is also called the th moment about the mean. Given , its standardized random variable is . Then the th standardized moment is .
The mean of is the first raw moment . The variance of is the second central moment . It is equivalent to . In words, the variance is the second raw moment subtracting the square of the mean.
The skewness of is the third standardized moment of . The kurtosis is the fourth standardized moment of . The excess kurtosis is the kurtosis subtracting 3. They are:
The calculation is usually done by expanding the expression inside the expectation. As a result, the calculation would then be a function of the individual raw moments up to the third or fourth order.
Note that the last line in skewness is a version that depends on the mean, the variance and the third raw moment. The calculation is illustrated with some examples. Practice problems are given in subsequent posts.
Example 1
Losses are modeled by a distribution that has the following density function. Calculate the mean, variance, skewness and kurtosis of the loss distribution.
The following shows the calculation of the first four raw moments.
The following shows the results.
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2017 – Dan Ma
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