a) Suppose that X₁, X₂, X3,...,X₁ is a random sample of n from population X which has a chi- square distribution with one degrees of freedom. (i) Find lim MX(t). 1400 (ii) Use the Central Limit Theorem to compute P(0.5 < X < 1.5), for n = 30. (iii) 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).3. Suppose X₁,..., Xn are iid with mean and variance o2 <∞o. Let Xn be the sample mean. (a) State the definition of convergence in distribution of a sequence of random variables. (b) Find the limiting distribution of Xn. (c) What is the limiting distribution of the random variable √n(Xn-u)/o? State the central limit theorem. (7 n)//n. Show6
- a) Suppose that X₁, X₂, X3,...,X, is a random sample of n from population X which has a c square distribution with one degrees of freedom. (1) Find lim Mg(t). 72-00Suppose 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).3) Given the joint distribution f (x1, x2) = a x,X2 for 0 < x, < 1, 0 < x2 < 1, and 0 elsewhere , find a possible value for a first. Then determine one number b randomly in any meaningful way you like. (b + 0) and calculate i)P(bX, +bX, < 1) ii )E(X, | X, = 1) - (E(X¡ | X, = 0.5) %D Make sure to write down all the details that you actually went through while solving this exercise in a very clear manner.
- 5. Does f(x) = In(x) satisfy the hypotheses of the Mean Value Theorem on [1,4]? If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem.20. Let X1, X2, ables, and set X, be independent, Exp(a)-distributed random vari- Y1 = X(1) and Y = X(k) – X(k-1); for 2Let 0 be the maximum likelihood estimator of the parameter 0. We are given that lim, 00 E(0) = 0 and 1. Then what is the asymptotic distribution of 0? %3D CRLB(@) Var(0) lim 00 O a. @~N(0,CRLB(0)). O b. ê- N(0, CRLB@) Oc O~ N(0,CRLB(0)). O d. - N(ô. CRLB(@).5Consider an infinite-server queueing system to which customers arrive according to a PP() (A > 0). Each customer's service time is exponentially distributed with rate u (u > 0). Since there are infinitely many servers, all customers begin their service immediately upon arrival (i.e., there is no queueing delay). Let X(t) be the number of customers in the system at time t. (a) Derive the limiting distribution (p) of {X(t) : t > 0}.b) 2 / 3 100% + (ii) Use the Central Limit Theorem to compute P(0.5 < X < 1.5), for n = 30. (iii) Use the Chebychev inequality to find the value of k for P(0.5 < X < 1.5), if n = 30. Let X₁, X₂, X3,...,Xn be a random sample of n from population X distributed with the f probability density function: 1 f(x; 0)=√2π0 0, x² e 20 if -∞Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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