Trong trường hợp này phân phối lognormal có thể cung cấp một xấp xỉ tốt, mặc dù không có lý thuyết để hỗ trợ sự lựa chọn dụ 7,8 (Ví dụ 7,7 tiếp tục) Giả sử, ngoài ra, được sự mất mát cá nhân được phân phối theo cấp số nhân vớiSau đó, sự phân bố của sự mất mát tối đa cho một khoảng thời gian năm k có dfVí dụ 7,9 | THE ROLE OF PARAMETERS 75 Example Demonstrate that the exponential distribution is a scale distribution. The distribution function of the exponential distribution is Fx x l-e-x e X 0. . Let Y cX where c 0. Then FY y Pr y y Pr cX y Prfx X c i _ e-v y 0. This is an exponential distribution with parameter c9. So the form of the distribution has not changed only the parameter value. Definition For random variables with nonnegative support a scale parameter is a parameter for a scale distribution that meets two conditions. First when the random variable of a member of the scale distribution is multiplied by a positive constant the parameter is multiplied by the same constant. Second when the random variable of a member of the scale distribution is multiplied by a positive constant all other parameters are unchanged. Example Demonstrate that the gamma distribution has a scale parameter. Let X have the gamma distribution and Y cX. Then using the incomplete gamma notation given in Appendix A Fy y Pr x - r fl. y indicating that Y has a gamma distribution with parameters a and CÔ. Therefore the parameter 9 is a scale parameter. It is often possible to recognize a scale parameter from looking at the distribution or density function. In particular the distribution function would have X always appear together with the scale parameter 9 as X 0. Finite mixture distributions Distributions that are finite mixtures have distributions that are weighted averages of other distribution functions. 76 MODELS FOR THE SIZE OF LOSSES CONTINUOUS DISTRIBUTIONS Definition A random variable Y is a k-point mixture2 of the random variables Xi Xỉ . Xk if its cdf is given by Fyfy aiFxffy a2 x2 H-------1 akFxk y 4-3 where all aj 0 and ai 2 Ok 1. This essentially assigns weight aj to the jth distribution. The weights are usually considered as parameters. Thus the total number of parameters is the sum of the parameters on the k distributions plus k 1. Note that if we have 20 .
