Suppose that X₁, X₂, X3 X is a random sample of n from population X which has a chi- square distribution with one degrees of freedom. (i) Find lim Mg(t). 819 (ii) Use the Central Limit Theorem to compute P(0.5<<1.5), for n = 30. (ii) Use the Chebychev inequality to find the value of k for P(0.5 < X < 1.5), if n = 30.
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- (i) Let X ~ Poisson(A). Then, using Chebyshev's inequality, show that P(X > 21) < 1/A. (ii) Suppose that the number of errors per computer program has a Pois- son distribution with mean 5. We get 125 programs. Let X1, X2, ..., X125 be the number of errors in the programs. Then, using the central limit theorem, find an approximate value for P(Xn < 5.5).An actuary models loss amounts under a insurance policy using the Pareto distribu- tion. The cdf of the Pareto distribution Suppose that 0 = 1400 and a = 1.6. What is the expected is F(x) = 1 (0/(x+0))ª. payment under a claim given that the policy has a deductible of 350 and a limit of 1300?7. a) Suppose that X is a uniform continuous random variable where 0Suppose that the claim size distribution of an insurance portfolio follows a Pareto distribution of the form α+1 α f(x) β = B\B+x (i) Derive a formula for the rth moment, ar, of this Pareto distribution in terms of its (r-1) th moment, αr-1. Show your steps clearly with reasons. (ii) From this expression find α3 and α4 using the known result for the mean μ of the above Pareto distribution. (You may assume that α>r, where a is one of the two parameters of the Pareto distribution).11 Assume the GPAS of high school students, X, follows a continuous uniform on the domain 1Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON