Q1: Prove(a) Markov's Inequality for non-negative continuous random variables U.(b) Chebeshey's Inequality for continuous random variables W, without using Markov's inequality.

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Answered: Q1: Prove (a) Markov's Inequality for…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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We say that a mean-zero random variable X is sub-Gaussian with parameter o² if the m.g.f. of X satisfies, for all t ER, Mx(t) ≤e ²0¹2 Let X be a sub-Gaussian random variable with parameter o². Using Markov's inequality, show that, for all a > 0, P(X ≥ a) ≤ e¯20² (Hint: the method of proof of the one-sided Chebyshev inequality may provide some in- spiration.)
1. (Markov's and Chebyshev's Inequality) a) Use Markov's inequality to show that for a sequence of positive random variables X1, X2, ... with values in N = {0,1, 2, ...} and lim;00 E[X;] = 0, it holds that lim;» P[X; = 0] = 1. b) Use Markov's and Chebyshev's inequality to give two upper bounds on the probability that an exponential distributed random variable with parameter A (rate) is larger than x. Calculate the bounds explicitly for ) →00 5 and x = 3,
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[stochastic process1]
• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain starts in state x, Pμ(A) the probability when the initial state is random with distribution μ. Ty = min{n ≥ 1: X₂ = y} is the first time after 0 that the chain visits state y. Px,y = Px(Ty < ∞) . Ny is the number of visits to state y after time 0. 4. Let Sn, n20 be a random walk on Z with step distribution 1-p 2 P 1) = 1/², P(X₁ = 1) = 2/₁ 2' 2' P(X₂ = 0) P(X₂ = − 1) = for some 0 < p < 1, p ‡ / . We may denote q=1 - p. That is, the increments (X₂)₁ are i.i.d. and S₂ = X₁ + … + Xn for n ≥ 1 and S = 0. Compute E[S2+1 Sn] for n ≥ 1. (b) Show that Mn = (q/p) Sn defines a martingale (with respect to (Xk)k-1). (c) Does the limit limn→∞ Mn exist almost surely? If yes, give a justification. If your answer is no, explain why. (d) Let T be the first time that S is equal to either −3 or 3. Compute P(ST = 3). Hint: You may use, without proof, the fact P(T<∞) = 1 and that the Optional Stopping Theorem…
5. Explain what is meant by BLUE estimates and the Gauss-Markov theorem. Mathematics relevant proofs will be rewarded. [:
Which statements are true? Select one or more: a. Markov’s inequality is only useful if I am interested in that X is larger than its expectation. b. Chebyshev’s inequality gives better bounds than Markov’s inequality. c. Markov’s inequality is easier to use. d. One can prove Chebyshev’s inequality using Markov’s inequality with (X−E(X))2.
Let X(1), X(2), ..., X(n) be the order statistics of a set of n independent uniform (0, 1) random variables.Find the conditional distribution of X(n) given that X(1) = s(1), X(2) = s(2), ..., X(n-1)=s(n-1).
Which of the following best describes the long-run probabilities of a Markov chain {Xn: n = 0, 1, 2, ...}? O the probabilities of eventually returning to a state having previously been in that state O the fraction of time the states are repeated on the next step O the fraction of the time being in the various states in the long run O the probabilities of starting in the various states • Previous Not saved Simpfun

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Q1: Prove (a) Markov's Inequality for non-negative continuous random variables U. (b) Chebeshey's Inequality for continuous random variables W, without using Markov's inequality.