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Tptaolsteh SET iii—L 1% (CAS Nov 05) Claim size, X, follows a Pareto distribution with parameters or and 6. A transformed distribution, Y, is created such that Y = X 1/ T .
Which of the foilowing is the probability density function of Y? 119ng1 a6“ry”“ 0&9 mfg/6)? :26“
A) (y+3)7+1 B) {y7+g)o+i C} (314.0)“: D) y[1+'(y7‘pj“r']a' +1 E) (yr+aJa+l 1% (CAS May 06) The aggregate losses of Eiffel Auto Insurance are denoted in euro currency
and follow a Lognormal distribution with ,u = 8 and 0' == 2. Given that 1 euro = 1.3 dollars,
which set of lognormal parameters describes the distribution of Eiffel's losses in dollars? A) p. n 6.15, a 2 2.26 B) n = 174,0 2 2.00 C) a = 8.00, ct = 2.60? D) a m 8.26, U = 2.00 E) ,u m 10.40, cr = 2.00 3 id}. (GAS May 06) Calculate the skewness ofa Pareto distribution with ct = 4. and 6 = l, 000 . A) Less than 2 B) At least 2, but less than 4 C At feast 4, but l th 6
D) At least 6, but less than 8 B) At 18 ast 8 ) ess an s. The distribution of a loss, X, is a twopoint mixture: (i) With probability 0.8, X has a twoparameter Pareto distribution with o: = 2 and 9 = 100.
(ii) With probability 0.2, X has a two—parameter Pareto distribution with o: = 4 and 6 — 3000.
Calculate Pr(X _<_ 200) . more B)0.79 (3)0.82 moss Boss 5, (SOA) The random variable N has a mixed distribution: (i) With probability p, N has a binomial distribution with q = 0.5 and m = 2. (ii) With probability 1 ~— 33, N has a binomial distribution with q = 0.5 and m = 4.
Which of the following is a correct expression for Prob(N = 2)? A) 0.125;)2 B) 0.375 + 0.125;) C) 0.375 + 0.125132 D) 0.375 — 0.125392 E) 0.375 « 0.12539 . . . . . 2
Q. Y is the mixture of an exponential random varlable With mean 1 and muting weight 5, and an exponential distribution with mean 2 and mixing weight Find the pdf, cdf, mean, variance and 90th percentile of Y. ‘?. A portfolio of insurance policies consists of two types of policies. Losses on Type 1 policies have a Pareto distribution with parameters a = 2, 6 z 100. Losses on policies of Type 2 have
an Inverse Pareto distribution with parameters 7' = 2 , 6 m 100. The policies are evenly divided
between the two types. A policy is chosen at random from the portfolio. Show that the distribution of the loss on the randomly chosen policy is a Pareto distribution and
identify the parameters. . An insurer selects risks ﬁ‘om a population that consists of three independent groups.
' The claims generation process for each group is Poisson.
o The ﬁrst group consists of 50% of the population. These individuals are expected to generate one claim per year.
° The second group consists of 35% of the population. These individuals are expected to generate tWo claims per year.
° Individuals in the third group are expected to generate three claims per year.
An individual is chosen at random from the population, and the individual is observed until a claim occurs.
(a) The number of full years with no claims until the ﬁrst claim year is denoted Y. Find E (13) Find the expected number of full years with no claims until the first claim year for each
separate group. Find the weighted average of those 3 numbers. 9. X. has a uniform distribution on the interval [0, 2"] for i = 1,2 3 3 3... Y. is amixture ofX1,X2,X3, with mixing weights c1 2 3?, a2 = , as m 3—
Fmd the mean and variance on and F341). 3 ...
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 Spring '11
 GABORLUKACS
 Calculus, Normal Distribution, Variance, Probability theory, probability density function, insurance policies, Pareto distribution

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