For some > 1, let X be a random variable taking the value 0 anelt, with probabilitic P(X-0)-1- P(X-1)- Var(X)-t Var(X)-1

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For Some 1 Let X Be A Random Variable Taking The Value 0 Anelt With Probabilitic P X 0 1 P X 1 Var X T Var X 1 1 (80.27 KiB) Viewed 33 times

For some > 1, let X be a random variable taking the value 0 anelt, with probabilitic P(X-0)-1- P(X-1)- Var(X)-t Var(X)-1/ E|X-1 DX-² None of these swers are correct □ Var(X)- O EXO EX-1/e ■Var(X)--1 Page: 4/ HOMEWORK +4 PROBABILITY AND STATISTICS ‒‒‒‒‒‒‒▬▬▬▬▬▬▬ ▬▬▬▬▬ +2/5/49+ Question & We still keep the notations of Question. For every a in R, define the probability 18) P(|X1|>a). Chebyshe's inequality provides an upper bound of 12¹. Besides, when one can com- pute exactly 18, we can compate the difference between the upper bound obtained by Chebyshe's inequality and. We call it the Chebyshev gap for X, at a and fort. We derate it (a). - Hence, it is easy to see that, ift-10 and a-8, thm 1/10 and that Chebyshe's inequality provides 9/64, an an upper bound of Hence, the Chebyshev gap for X, ata-8 and for t-10 in alto 9/64-1/10-9/80. In other words, one can write (8) 0.04 O (6) 04 -7/12 1-1/26 3 0% ☐ - TO 0-12/7. -1/10. ο Mars of these arum ETA ME correct Question 9.4 We still keep the notations of Question to. One can affirm that: None of the wes are correct D-0 O iny -2/9 (90) -1/3 (10) 4101 (10)-100 O (10) (10) Question 10. We still keep the notations of Question to||. More generally, for way t in (1,00), one can affirm that: -0, Va -0, VL. Nome of these we are correct HOMEWORK +4 PROBABILITY AND STATISTICS Page: 5/ HOMEWORK 4 -1/10 ▪‒‒‒‒‒‒‒‒‒‒ —‒‒‒‒ Question 11. We still keep the notations of Question to. More generally, for any tin (1,00), one can affirm using Chebychev's inequality, that l 18 Vaat Va None of these answers are correct “One might have to use the inequality : – 1 € a valid for all sən. Question 12.4 We still keep the notations of Question || to. One wonders here if one can find a gep xmaller than a certain number One can affirm that: One can find a gap smaller than 10-2 None of these awers are correct One can find a gap smaller than 10, for all in Nº. □ One can find a gap muller than 10 There are some pin N*, for which one can not find a gap smaller than 10-³, for all p in Nº. One can not find a gap analler than 10-7. Question 13. (Optional) We still keep the notations of Question to. One can affirm that: None of the recom □ One can not improve Chebychev Inequality, in general. □ One can improve Chebychaw Inequality, in general PROBABILITY AND STATISTICS +2/6/48+ Page: 6/ ▪▪▪▪ +2/7/47+ Question 14.4 Let X1, X₂, be a sequence of independent random variables, with E|X₁| - and Var(X₁)-for-1,2,.... Suppose that 0 < o} < M, for all i. Let a be an arbitrary ponitive mumbur: Cree canalfirm that: None of them arowers are correct O -2 ▪ P Var(X1) +--- + Var(X₂) P (T-7¹->) < Var(X) +-_-_-4 Var(X₂) •(-=-=-2 - > -) < □P(->20) <Var(X1) +++ Var(Xn) = P(x=-=> ) <2V(X) +---+ Var(X₂) +---+' OP