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44721 (C) If you wish to do the exercise directly: differentiate. We don't need the denominator of the derivative, since all we want to do is set the entire expression equal to zero. ll. 12. 14/9,. (A) Using the means and coefficients of variations, we have Var(A) =Var(B) = 25 Var(C)= 100 Also Cov( (A+ B), (A+ C)) = Cov(A,A)+ Cov(A, C)+ Cov( B,A)+ Cov(B, C)= Var(A)+ 0 + 0 + 0 = 25 because A, B, and C are independent. 13. For the inverse exponential distribution, the mode is~~ so(}= 20,000. 5. 14.

2. [4B-S90:37] (2 points) Liability claim severity follows a Pareto distribution with a mean of 25,000 and parameter a= 3. If inflation increases all claims by 20%, the probability of a claim exceeding 100,000 increases by what amount? 3. (4B-F97:26] (3 points) You are given the following: In 1996, losses follow a lognormal distribution with parameters J1 and a. • • In 1997, losses follow a lognormal distribution with parameters J1 +Ink and a, where k is greater than 1. • In 1996, lOOp% of the losses exceed the mean of the losses in 1997.

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.