28. Show {Xn, n ≥ 1} are independent if o(X₁,..., Xn-1) || σ(Xn) are independent for each n ≥ 2.

Q: 49. Prove that x +> 2 (x1/2 – x-1/2)2. 2 for all x > 0. Hint: Consider

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Q: (a) Show that if n € N, then n+1 In dx. I

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Q: 1 Q6) Let H(x):=k for x=, (k eN) and H(x):=0 elsewhere on [0,1] k Determine whether H(x) eR[0,1 or…

A: Unbounded function is not Riemann integrable.

Q: Show that for all x>0, e* >1+x+÷ 2.

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Q: Show that it is not true that for every error 1/m there exists n such that |Ek=1 polt – r/(1 - r)| <…

A: We will show that for every m there does not exist any unique n that works for all r in (0,1)

Q: 2. Prove that |µ(n)| _ S(s) S(2s) n=1 if s > 1.

A: Riemann zeta function is a complex-valued function defined as ςs=∑n=1∞1ns.  There is a relationship…

Q: Let Sy := {1-(-1):NEN. Find iNf Sy and sup Sy

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Q: Let: A = {n € Z|0< n < 3} B = {Vn|n €Nand 1 <n < 4}

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Q: Define a function S: Z+ → Z+ as follows: For each positive integer n, S(n) = the sum of the positive…

A: Solution

Q: A container contains 35 green tokens, 5 blue tokens, and 3 red tokens. Two tokens are randomly…

A: There are total (35+5)=40 non-red tokens and 3 red tokens in the container.  And total (35+5+3)=43…

Q: Let X-N(65,36) P(x> 55) = O 0.5232 none O 0.9525 0.98077

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Q: 15. Let z = 2cos0 + i(1 – 2sin0) where -n < 0 < n Show that Iz – il = 2

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Q: 3. Let h: Z→ Z be the function h(r) = = x mod 4. Find h-¹(0), h-¹({0, 1, 2, 3}), and h-¹(4). Explain…

A: In this question we can easily observe that that output of the function is the remainder when any…

Q: 1.20 Show that, if 0 < |7₂| < |x₁] then, for all n ≥ 1, |T1| |I₁|-|1₂|| n 12 I in-1

A: Given that Let, Now consider geometric series As

Q: Let X1 ∼ N(0.7,1.4), X2 ∼ N(−0.5,1.2), X3 ∼ N(0.9,1.7), and X4 ∼ N(−0.2,2.1), independently. 1.…

A: X1 ∼ N(0.7,1.4), X2 ∼ N(−0.5,1.2), X3 ∼ N(0.9,1.7), and X4 ∼ N(−0.2,2.1), independently.

Q: Prove that T S. Prove that S £ T.

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Q: Let E = . Determine the smallest choice of N > 0 such that N. |n(n+1)|

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Q: Show that φ(n2) = nφ(n) for every n ∈ Z+.

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Q: 4. Prove that if (an) is increasing and an ̸= 0 for all n ∈ N then ? 1/ an is decreasing.

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Q: Let x = [1, -3, 2]T. Find ||x||₁, ||x|4, and ||×||∞·

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