20.21. Suppose that the sequence (X) is fundamental in probability in the sense thatfor e positive there exists an N such that P[IX-X|>e] N.(a) Prove there is a subsequence (X) and a random variable X such thatlim XX with probability 1. Hint: Choose increasing n such thatP[IX-X₁>2-k] <2-k for m, n ≥n. Analyze P[X-Xl>2-k].→PX.(b) Show that Xn

Given: PXm-Xn>∈<∈ for m,n >Ne

Answered: 20.21. Suppose that the sequence (X) is…

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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be a sequence of iiid. cntinmous --.. distribution r.V. Let N= min {n: X2 X, z Xz Z -.. 2 Xn-l , Xawi < Xn}. Find E CN] [ Hint:ECN] as Compute
10. Let {an}, {bn} and {cn} be sequences of real numbers. Which of the following statements are true? I. If {an} is convergent and {bn} is bounded, then {a„bn} is convergent. II. If lim an = L and lim b, = 0, then n-00 lim anbn = 00. n-00 III. If a, #0 for all n e Z+ and lim n→∞ An 0, then %3D lim an = 00. n-00 IV. If lim an = 0 and {bn} is bounded, then n-00 lim anbn = 0. n00 V. If b, < an < Cn for all n e Zt, {bn} and {cn} are convergent, then {an} is also convergent. VI. If {an} is unbounded, then {an} is divergent. (a) I, II (b) I, IV (c) IV, VI (а) , II, V (е) 1, IV, V
5. Show that the following sequences of sets, {Ck }, are non-decreasing, (1.2.16), then find limä-∞ Ck- Ck = {x : 1/k < x < 3 – 1/k}, k = 1, 2, 3, ....
A jar contains n red balls labeled from 1 to n where n ≥ 2 and m blue balls. a. Two balls are selected at random, one after another, from the jar without replacement. What is the probability that the pair of ball is red? b. One red ball is selected at random from the jar, let pn be the probability of selecting a red ball labeled with a prime even number. Find pn and then compute lim Pn. n-∞
2. Suppose the sequence {sn} satisfies |8n+1 = 8n| ≤ 72²2 for all n E N. Prove that {n} converges.
2- Suppose that (xn) is a sequence such that xn > 0 for all NE N. 1 = 0. Show that; Xn → 0 as n → 0∞ lim n→o Xn O lin I0 - Show also; Xns-0∞ as n
23. Suppose (An) are events. (a) If for each k show (d) Show iff n-1 Σ P(An! ΠA) = 00, n=k i=k (b) What is the relevance to the Borel Zero-One Law? (c) Is it enough to assume P(lim sup An) = 1. 8x 8 n=1 n- P(A₂A) = ∞o? i=1 P (lim sup An) = 1 81x 8 Σ P(AAn) = 00 n=1 for all events A such that P(A) > 0.
2. Show that for all n ≥ k ≥ 0, 3. Show that for all n ≥ k ≥ 0, a. town b. rinse c. print d. carry n (7²) = (₂₁ n e. expect f. anticipation n+1) n (" + ¹) = (^) + (²₁). Hint. Start with the right-hand side and simplify. Note that k! = k· (k − 1)!. 4. How many different letter arrangements can be made from the following words? n 1 - k 129 Like
1) need a and b
5
6. Let {n} and {yn} be the following sequences that repeat in cycles of four: n = 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, ... and Find (a) lim inf xn + lim inf Yn Yn = 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2... (b) lim inf(xn + Yn) (c) lim sup n + lim sup yn
Sn, 4. Let Sn cos(2). Use the definition to find lim supsn and lim inf 3 then explain why sn has no limit. (hint Thm 10.7) =

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