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Let H(x) be the cumulative hazard function of X. Calculate H(3). 2. The random variable X has a uniform distribution on (0, 1]. Let h(x) be its hazard rate function. 75). C/4 Study Manual-!! 0 ASM Exercises continue on the next page ... 1: Summary of Probability Concepts • Median is 50th percentile; nth quartile is 25nth percentile. Probability Functions F(x) = Pr(X S x) S(x) = 1- F(x) • Mode is x which maximizes f(x). f(x)= dF(x) dx H(x)=-lnS(x) • Af;l(O)= E[X11 ], whereMinJ is the h(x)= dH(x) dx n!

You are not responsible for graphs, but they may help you understand the distributions. d 1 x2:u { You recognize a uniform distribution both by its finite support and by the lack of an x in the density function. Its moments are - d+u E [X) - - - 2 Var(Xf= (u- d)2 12 Its mean, median, and midrange are equal. The best way to calculate the second moment is to add up the variance and the square of the mean. However, if d = 0, then E[X2) = u 2 /3. The uniform distribution is not directly in the tables, so I recommend you memorize the formulas for mean and variance.

1. 2 Sk(X) = Sk(2X) 2. -Sk(Y)= Sk(-Y) 3. ISk(X)I ~ ISk(X + Y)l (A) 2 (B) 3 (C) 1,2 (D) 2,3 (E) None of A, B, C, or D l ' C/ 4 Study Manual- II th edition Copyright ©2010 ASM Exercises continue on the next page ... ,. 8. [4B-S97:21] (2 points) You are given the following: • Both the mean and the coefficient of variation of a particular distribution are 2. • The third moment ofthis distribution about the origin is 136. Determine the skewness of this distribution. Hint: The skewness of a distribution is defined to be the third central moment divided by the cube of the standard deviation.

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A.S.M. study manual for Exam C/exam 4 : construction and evaluation of actuarial models by by Abraham Weishaus.

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