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Since the median is 800, ell= 800 and J1. = ln800. Since the mean is 1000, eJL+a2f 2 = 1000. 23. By the Law of Total Probability, the probability that the sum of two claims is greater than 20 is the integral over all x of the probability that the sum is greater than 20 given that Kira's claim is x times the density of Kira's distribution. ) If X is Kira's claim and Y is Kevin's claim, then Pr(X + Y > 20)= ioo Pr(X + Y > 20 I x)f(x)dx = i oo 1 Pr(X+Y>20jx)-e-xfl 2dx 12 0 Now, Pr(X + Y > 20) = 1 if X> 20 since Y can't be negative.

And a. P(x) is the standard normal distribution function, for which you are given tables. Scr 2 E[X2] = e2J-L+2cr 2 More generally, E[Xk] = E[ekY] = Mv(k), where Mv(k) is the moment generating function of the corresponding normal distribution. 6 graphs three lognormals with mean 50. - a 2 ), as stated in the tables. = 2, the mode is off the graph. As a gets lower, the distribution flattens out. 1 is a summary of the forms of probability density functions for common distributions. 4 which is on the syllabus, is background for something we'll learn later in credibility.

1 Scaling A parametric distribution is a member of a scale family if any positive multiple of the random variable has the same form. In other words, the distribution function of eX, for c a positive constant, is of the same form as the distribution function of X, but with different values for the parameters. Sometimes the distribution can be parametrized in such a way that only one parameter of eX has a value different from the parameters of X. If the distribution is parametrized in this fashion, so that the only parameter of eX having a different value from X is (}, and the value of() for eX is c times the value of(} for X, then () is called a scale parameter.

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