Let X Exponential (3)a) Find the Markov upper bound for P(X>10).b) Find the Chebyshev upper bound for P(X>10). State which of the two bounds found is better.c) Find the Chernoff upper bound for P(X>10). Clearly state the allowable range for t.d) Let Zn be a RV from a Gamma(n, 3) distribution for n=1,2,... . What doesconverge in probabilityto? Please state both the value and the reasoning by which Z converges in probability to it.n

Markov's inequality provides an upper bound on the probability that the a non negative randm…

Answered: Let X Exponential (3) a) Find the…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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How many words can be formed by permutation of the letters of the word DEPRESSION such that each obtained word does not contain 2 consecutive identical letters? (Hint: use the principle of inclusion and exclusion)
Consider a (non)-symmetric random walk M = Xo +X1+X2+…+Xn where X;'s (i> 1) are independent and identical distributed, P(X; = 1) = p and P(X;=-1) =1–p and Xo = 0. Let Sn = eaS,-n Denote Fn= 0(X1,...,Xn). Find a such that (Sn)n20 is a (Fn)n20 - martingale.
Suppose that independent Bernoulli trials with parameter p are performed successively. Let N be the number of trials until x successes, and X be the number of successes in the first n trials. Show that P(N = n) = "P(X = x).
• 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…
Pedestrians approach a crossing from the left and right sides following independent Poisson processes withaverage arrival rates of λL = 5 and λR = 1 arrivals per minute. Each pedestrian then waits until a lightis flashed, at which time all waiting pedestrians must cross to the opposite side (either from left to right orfrom right to left). Assume that the left and right arrival processes are independent, that the light flashesevery T = 2 minutes, and that crossing takes zero time – it is instantaneous.1. What is the probability that in a particular crossing, there are total 10 pedestrian and they are allcrossing from left to right?
Consider a continuous-time Markov chain whose jump chain is a random walk with reflecting barriers 0 and m where po,1 = 1 and pm,m-1 =1 and pii-1 = Pii+1 = for 1
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.
(6) Give an example of a Markov chain Xn on a countable state space S and a function g on S such that g(Xn) is not a Markov chain. Can you give any conditions on Xn and g which ensure that g(Xn) is a Markov chain?
Let X1, X2,..., Xn be independent exponential variables, parameter λ. Show by induction that X2 S = X₁ + X₂ + + Xn has the r(λ, n) distribution.
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Let X be a Markov chain on S, and let I: S" {0, 1}. Show that the distribution of Xn, Xn+1,..., conditional on {I (X₁,..., Xn) = 1} n {Xn = i}, is identical to the distribution of Xn, Xn+1,... conditional on {Xn = i}.

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