a) 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 Mx(t). 11-00 (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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- 3. A random variable X has a cumulative distribution function if 0 0.5). (e) Find P(X < 1.25). (f) Find P(X = 1.25).(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).Let X1,..., Xn be independent and identically distributed random variables with Var(X1) < ∞. Show that n 1 EjX;→p EX1. п(п + 1) j= Note. A simple way to solve a problem of showing Yn →p a is to establish limn EYn = a and limn Var(Yn) = 0.
- 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. ShowNumber 21Find the value of ( K) in t-test such that P( K< T < -1.761)=0.045 for a random sample of size
- 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).2. A random variable X has a p.d.f. f(x) given by √(1-x)6 03) 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.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 22. Consider a random sample X₁, X2,..., Xn of size n from the Uniform (0, 0) distri- bution, where @ > 0. (i) Show that the p.d.f. q(t; 0) of the sufficient statistic is given by T(X)= max X₁ 1SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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