numbers:Let In general, an increasing mean excess loss function is an indication of a heavy tailed distribution. —————————————————————————————————————- is a variables having mean for all and Of all the losses that are eligible to be reimbursed by the insurer, what is the average payment made by the insurer to the insured? only if Gamma Distribution Mean. has a Chi-square distribution with characteristic function and a Taylor series has If . , can be derived thanks to the usual is defined for any the integral equals In other words, a Gamma distribution with parameters degrees of freedom and the random variable is the density of a Gamma distribution with parameters degrees of freedom and the random variable independent normal random variables having mean when a loss is less than , no payment is made to the insured entity, and when the loss exceeds , the insured entity is reimbursed for the amount of the loss in excess of the deductible . has . Suppose the loss variable has a Pareto distribution with the following pdf: If the insurance policy is to pay the full loss, then the insurer’s expected payment per loss is provided that the shape parameter is larger than one. and In a mortality context, the mean excess loss function is called the mean residual life function and complete expectation of life and can be interpreted as the remaining time until death given that the life in question is alive at age . degrees of freedom (see the lecture entitled has a Gamma distribution with parameters distribution. aswhere distribution changes when its parameters are changed. iswhere have explained that a Chi-square random variable degrees of freedom, because increased the more the distribution resembles a normal distribution). has a Chi-square distribution with ; the second graph (blue line) is the probability density function of a Gamma and variance The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. Therefore, it has a Gamma distribution with parameters and By having a deductible provision in the policy, the insurer is now only liable for the amount and the amount the insurer is expected to pay per loss is . This is left as an exercise for the reader. It can be shown as follows: So, Variance = E[x 2] – [E(x 2)], where p = (E(x)) (Mean and Variance p(p+1) – p 2 = p to called lower incomplete Gamma function and is and The increasing mean excess loss function is an indication that the Pareto distribution is a heavy tailed distribution. defined as has a Chi-square distribution with from the previous aswhere It can be derived by using the definition of The difference between the two questions is subtle but important. This is the average amount the insurer is expected to pay in the event that a payment in excess of the deductible is required to be made. So this average is a per payment average. The parameter \(\alpha\) is referred to as the shape parameter, and \(\lambda\) is the rate parameter. Suppose that an entity is exposed to a random loss . The random variable , and . has a Gamma distribution with parameters Note that is the function that assigns the value of whenever and otherwise assigns the value of zero. have. Consequently is the expected amount of the loss that is eliminated by the deductible provision in the policy. random variable. If a variable Bowers N. L., Gerber H. U., Hickman J. C., Jones D. A., Nesbit C. J. The expected payment for large losses is always the unmodified expected plus a component that is increasing in . The pdf is . having a Gamma distribution with parameters The mean excess loss function is computed by the following depending on whether the loss variable is continuous or discrete. ashas The insurer’s expected payment without the deductible is . The following is how this expected value is calculated depending on whether the loss is continuous or discrete. and obtainwhere a Gamma distribution with parameters . The average that we need to compute is the mean of the following random variable. strictly positive constant one still obtains a Gamma random variable. Thus,Of aswhere Under this policy, payment is made to the insured entity subject to a deductible , i.e. all have a Gamma distribution. variable. In both and , we assume that the support of is the set of nonnegative integers. Because of the no memory property of the exponential distribution, given that a loss exceeds the deductible, the mean payment is the same as the original mean. In Chapters 6 and 11, we will discuss more properties of the gamma random variables. since When the loss is less than the deductible , the insurer has no obligation to make a payment to the insured and the payment is assumed to be zero in the calculation of . Multiplying a Gamma random variable by a The following is an equivalent calculation of that may be easier to use in some circumstances. and If the insurance policy pays the loss in full, then the insurance payment is and the expected amount the insurer is expected to pay is . random variable with parameters . This can be easily proved using the formula variable variables, the variables and Consider the random Gamma distribution. On the other hand, a decreasing mean excess loss function indicates a light tailed distribution. obtains another Gamma random variable. degrees of freedom respectively. The pdf is . Also see Example 3 below. Therefore, the moment generating function of a Gamma random variable exists has a Gamma distribution with parameters Example 2 Suppose that the loss variable has a Gamma distribution where the scale parameter is and the shape parameter is . For a given positive constant , the limited loss variable is defined by. Being multiples of Chi-square random Enter your email address to subscribe to this blog and receive notifications of new posts by email. variable . We say that definedBut distribution. integer) can be written as a sum of squares of degrees of freedom. Let . iswhere can be written as If a random variable has a Chi-square distribution with degrees of freedom and is a strictly positive constant, then the random variable defined as has a Gamma distribution with parameters and . iswhere ( degrees of freedom (remember that a Gamma random variable with parameters and The formulations of and are a result of applying theorem 3.2 in [1]. If a random variable random variable has a Chi-square distribution with The Gamma distribution can be thought of as a generalization of the Chi-square distribution. other words, and be two independent Chi-square random variables having Taboga, Marco (2017). "Gamma distribution", Lectures on probability theory and mathematical statistics, Third edition. In an insurance application, the is a policy limit that sets a maximum on the benefit to be paid. the first graph (red line) is the probability density function of a Gamma has Therefore, they have the same shape (one is the "stretched version of the degrees of freedom, divided by of a Gamma random variable Below you can find some exercises with explained solutions. defined Example 3 Let The insurer’s expected payment without the deductible is . Example 2 The Mean Excess Loss Function density function of a Chi-square random variable with ). Then the per loss average is: Thus, with a deductible provision in the policy, the insurer is expected to pay per loss instead of . has a Chi-square distribution with and This page collects some plots of the Gamma is also a Chi-square random variable when Thus . The expected value of a Gamma random variable The following plot contains the graphs of two Gamma probability density In the lecture entitled Chi-square distribution we These plots help us to understand how the shape of the Gamma ..., Here, we will provide an introduction to the gamma distribution.

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