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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2. Let the sequence (n) be recursively defined by x1 = √√2 and xn+1 = √2+xn, n≥1. Show that (xn) converges and evaluate its limit.
4. Suppose g(n) = Σdn f(d). Write these sums out explicitly for n = 1, 2, ..., 12. How can we reverse this, and express f in terms of the values of g ? To start, express each value ƒ(1), ƒ(2),..., ƒ(12) as a combination of values of g. What patterns do you observe? Any conjectures?
4. Consider a sequence aj = 3 and an+1 = v V12 + an for each n > 1. (a) by 4. Show by induction that {an} is monotone increasing and bounded above
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, ....
2. Suppose the sequence {sn} satisfies |8n+1 = 8n| ≤ 72²2 for all n E N. Prove that {n} converges.
1. Show the sequence an = (n+1)/(n-1) is strictly decreasing and bounded below, and give its limit.
3. Let an = (3n +4n)¹/n (a) Prove that for all n € N, 4 < an = 4(1 + (²)n)¹/n < 4.2¹/n (b) Prove that the sequence {an}=1 is convergent and find its limit.
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
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) =
(e) Find a closed formula for the sequence (Cn)neN defined as follows: 1 if n = 1, Cn = 1 if n = 2, 8. Cn-1 · (Cn-2)6 if n ≥ 3. Explain your answer. Hint: While the recurrence for (cn) provided here is not given in a form discussed earlier in the lectures, one can transform the problem into an equivalent one with a non-homogeneous linear recurrence. Try to find such transformation and then solve the obtained non-homogeneous recurrence.

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