20.21. Suppose that the sequence (X) is fundamental in probability in the sense that for 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 that lim XX with probability 1. Hint: Choose increasing n such that P[IX-X,>2-k]<2-k for m, nzn. Analyze P[IX-Xn>2-k]. (b) Show that X₁ →p X.

Q: 1 and for k 01let uk+1 be determined by the 4. 1 = 2uk – 2u. This sequence tends to a limit; Set uo…

A: Given: Let  u0 =14 and  uk+1 for k≥0 And, uk+1 =2uk -2uk2 ......1

Q: 4. Use the definition to show that the following sequences are Cauchy: (a) {3n+2}n=1 (b) {1 n=1 (c)…

A:

Q: 1. A fair die is tossed 450 times. Let X denote the number of times that a number <4 is obtained. a)…

A:

Q: A nonstationary respectively: random sequence X[n] has the following mean and autocovariance…

A: In this problem, we are given a non-stationary random sequence, X[n], with a specified mean sequence…

Q: a) Suppose (pn) has limit L, and let S be the set of values of the terms of (pn). Can we say that L…

A:

Q: S² P, o² and „", → 1, their product converges in *6. In lecture it is claimed that since "S2…

A: Given: Yn is a sequence of random variables that converges in probability to Y. So,…

Q: 5. Consider the sequence {an}. Defined by the recurrence relation; a₁ = 1, an+1 = (2an+3), for n ≥…

A: We have to prove the sequence is increasing bounded above by

Q: Consider the question: Apply five iterations of the Gauss-Seidel method, with initial approximation…

A:

Q: *? Let (sn) be a sequence that converges. (a) Show that if sn > a for all but finitely many n, then…

A:

Q: Let (sn) be a convergent sequence, and suppose lim sn > a. Prove there exists a number N such that n…

A:

Q: Let (an) be the sequence with an = (-1)” +212/2. (i) Use the fact that 1/n² → 0 as n → ∞ and…

A:

Q: 3. Assume the sequence of positive numbers (sn) increases monotonically to infinity. Prove that Sn+1…

A: Let Sn be a sequence of positive numbers sn increase monotonically to infinity. ⇒limn→∞sn=∞     ∵ sn…

Q: 20.21. Suppose that the sequence (X) is fundamental in probability in the sense that for e positive…

A: Given: PXm-Xn&gt;∈&lt;∈ for m,n &gt;Ne

Q: 3.15 let (fa) be a Sequence of function on [₁1] defined by fak+1 (x)= of xe lo1/3] otherwise, رع i…

A:

Q: 21. Suppose (An) is a sequence of events. (a) If P (An) → 1 as n → ∞, prove there exists a…

A:

Q: 5.80 Show that these facts are true. Use the definition of binomial coefficients to show that each…

A: We expand left hand side i.e. nCn nCn= n!/n!*(n-n)! We know that 0!= 1 so  nCn = n!/n! = 1 This…

Q: Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all but finitely many n, then lim…

A: Let (sn) be a convergent sequence, and assume that sn ≥ a for all but finitely many n.

Q: Prove that x ∈ E' iff there exists a sequence (xn) in E\{x} such that xn -> x.

A:

Q: 12. Prove the following facts about the Fibonacci sequence (Fn): (1) (Chapter 10, #26) E-1 F =…

A: Given, The Fibonacci sequence (Fn) Fn=Fn-1+Fn-2, where F0=0 &amp; F1=1

Q: If {f} converges in the mean squa

A:

Q: x[n]=(0,7)n.u[n] and h[n]=(0,9)n.u[n] to be y[n]=x[n]*h[n] Get the result of my processing…

A: Solution is in step 2

Q: 2. Suppose the sequence {sn} satisfies Sn+ 1 - Sn ≤ -1/24 n² For all ne N. Prove that {sn}…

A: Given that the sequence  satisfies .The objective is to find for all ,  converges.

Q: 3. Let {an be a sequence in R. (a) Show that it has a monotonic subsequence. (b) Prove that a…

A:

Q: 26. (Barndorff-Nielsen) Suppose {En} is a sequence of events such that lim_ P(En) = 0, ΣP(EnE+1) n.…

A: Given::- 26. (Barndorff-Nielsen) Suppose (En) is a sequence of events such that lim P(En) = 0,…

Q: ) Prove that if XnXnand YnYn are any two sequences of real numbers such that XnXn converges to xx…

A:

Q: Find the probability that a random permutation of [4k] has no cycle of length> 3k. Also, Hetermine…

A: In this problem, we are tasked with finding the probability that a random permutation of the set…

Q: 4. If (x) is a sequence in an inner product space X such that the series converges, show that (sn)…

A: It is given that xj is a sequence in an inner product space X such that the series x1+x2+⋯…

Q: 3. If X~B(6,0.2) , by using the binomial table, determine (a) P(X = 2) (c) P(X 5) (d) P(X < 4)

A: As per our policy we can solve only 3 sub parts per post. Kindly re-post for remaining questions.…

Q: 2) The purpose of this problem is to give you some practice with the most common application of The…

A:

Q: 1. Let o(n) = Edn d. Example: For primes p, q, a(p) = p+ 1. a(pq) = pq+ p+q+ 1. Show that o(n) = n…

A:

Q: (2) Σ(1+1) (b) 1 ΣΕ η k=1 2 /3k + 1 + V3k – 2 n k + 1 (Σ (k+2)! k=1 η (a) Π k=2 k3 - 1 - k³ +1

A: As per our company guidelines, we are advised to solve only the first three sub parts. Kindly repost…

Q: X = factorial(3)   def factorial(n): if n == 1: return 1 else: return n* factorial (n-1) def…

A: Definition of Mathematical Induction:       It is essentially used to prove that a statement P(n)…

Q: 4. If {Cn} is a convergent sequence of real numbers, does there necessarily exist R > 0 such that…

A: Given that cn is a convergent sequence of real numbers and let it converge to a limit l. Then by…

Q: Consider a sequence {X,,n = 1,2, 3, ·.} such that with probability Xn = 1 with probability a.s. Show…

A: @solution::::

Q: 14. Let {f} be a sequence on C(X) where X CR is compact. Assume that fn (x) ≥ fn+1(x) for all x E X.…

A: Given:A sequence of functions  in the space of complex-valued continuous functions , where  is a…

Q: uppose that the sequence x0, x1, x2... is defined by xo = 3, x₁ =0, x₂=5, and xk+3 = 13xk+1+12x for…

A:

Q: 24. Let S denote the multiset {o e₁,00 e2, ∞ ek}. Determine the exponential where ho 1 and,…

A: Let us write  the multiset first...and name it as S Then we will define g(x) Given h0=1 n&gt;=1…

Q: 2. Suppose that the sequence {an} converges to a and that a < 1. Prove that the sequence {(an)"}…

A: