Give a counterexample: a) Xn + Yn converges if and only if both Xn and Yn...

Give a counterexample:

a) Xn + Yn converges if and only if both Xn and Yn converge.

b) Xn Yn converges if and only if both Xn and Yn converge.

Expert Solution

Colby Messinger answered 2 years ago

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn...

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn ≤ lim inf(xn + yn) ≤ lim sup(xn + yn) ≤ lim sup xn + lim sup yn. Give example where all inequalities are strict. b)Let (zn) be the sequence defined recursively by z1 = z2 = 1, zn+2 = √ zn+1 + √ zn, n = 1, 2, . . . . Prove that (zn) is convergent and find its limit. Hint; argue...

4.2.1. Problem. Suppose that (xn) and (yn) are sequences of real numbers, that xn → a...

4.2.1. Problem. Suppose that (xn) and (yn) are sequences of real numbers, that xn → a and yn → b, and that c ∈ R. For the case where a and b are real numbers derive the following: (a) xn +yn →a+b, (b) xn −yn →a−b, (c) xnyn → ab, (d) cxn → ca, (e) xn/yn → a/b if b ̸= 0.

Prove that if {xn} converges to x does not equal to 0 and xn is non-zero...

Prove that if {xn} converges to x does not equal to 0 and xn is non-zero for all n, then there exists m > 0 so that |xn| ≥ m for all n.

1. . Let X1, . . . , Xn, Y1, . . . , Yn be...

1. . Let X1, . . . , Xn, Y1, . . . , Yn be mutually independent random variables, and Z = 1 n Pn i=1 XiYi . Suppose for each i ∈ {1, . . . , n}, Xi ∼ Bernoulli(p), Yi ∼ Binomial(n, p). What is Var[Z]? 2. There is a fair coin and a biased coin that flips heads with probability 1/4. You randomly pick one of the coins and flip it until you get a...

A sequence (xn) converges quadratically to x if there is some Q ∈ R such that...

A sequence (xn) converges quadratically to x if there is some Q ∈ R such that |xn − x| ≤ Q/n^2 for all n ∈ N. Prove directly that if (xn) converges quadratically, then it is also Cauchy.

Show that a monotone sequence converges if and only if it is bounded.

Consider a random sample (X1, Y1), (X2, Y2), . . . , (Xn, Yn) where Y...

Consider a random sample (X1, Y1), (X2, Y2), . . . , (Xn, Yn) where Y | X = x is modeled by Y=β0+β1x+ε, ε∼N(0,σ^2), where β0,β1and σ^2 are unknown. Let β1 denote the mle of β1. Derive V(βhat1).

Let X = {Xn:n= 0,1,...,} be a DTMC on state space S. Define Yn = (Xn,...

Let X = {Xn:n= 0,1,...,} be a DTMC on state space S. Define Yn = (Xn, Xn+1). Prove that Y = {Yn: n = 0,1,2,...} is a DTMC. Specify its state space and the transition matrix.

3. Let X1...Xn be N(μX,σ) and Y1...Yn be iid N(μy,σ) with the two samples X1...Xn, and...

3. Let X1...Xn be N(μX,σ) and Y1...Yn be iid N(μy,σ) with the two samples X1...Xn, and Y1...Xn independent of each other. Assume that the common population SD σ is known but the two means are not. Consider testing the hypothesis null: μx = μy vs alternative: μx ≠ μy. d. Assume σ=1 and n=20. How large must δ be for the size 0.01 test to have power at least 0.99? e. Assume σ=1and δ=0.2. How large must n be for...

3. Let X1...Xn be N(μX,σ) and Y1...Yn be iid N(μy,σ) with the two samples X1...Xn, and...

3. Let X1...Xn be N(μX,σ) and Y1...Yn be iid N(μy,σ) with the two samples X1...Xn, and Y1...Xn independent of each other. Assume that the common population SD σ is known but the two means are not. Consider testing the hypothesis null: μx = μy vs alternative: μx ≠ μy. a. Find the likelihood ratio test statistic Λ. Specify which MLEs you are using and how you plug them in.

Question

Give a counterexample: a) Xn + Yn converges if and only if both Xn and Yn...

Solutions

Expert Solution

Related Solutions

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn...

4.2.1. Problem. Suppose that (xn) and (yn) are sequences of real numbers, that xn → a...

Prove that if {xn} converges to x does not equal to 0 and xn is non-zero...

1. . Let X1, . . . , Xn, Y1, . . . , Yn be...

A sequence (xn) converges quadratically to x if there is some Q ∈ R such that...

Show that a monotone sequence converges if and only if it is bounded.

Consider a random sample (X1, Y1), (X2, Y2), . . . , (Xn, Yn) where Y...

Let X = {Xn:n= 0,1,...,} be a DTMC on state space S. Define Yn = (Xn,...

3. Let X1...Xn be N(μX,σ) and Y1...Yn be iid N(μy,σ) with the two samples X1...Xn, and...

3. Let X1...Xn be N(μX,σ) and Y1...Yn be iid N(μy,σ) with the two samples X1...Xn, and...