For n = 1,2..., let Xñ be a r.v., and let X be a r.v. Prove that if XnX then XX.

Q: For each n ≥ 1, take An= (-1/n, 1/n). Then, 1 An {0} and U1 An (-1, 1). Prove these.

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Q: Prove that for all a € R, 0 < r² + 2 VI

A: Prove the following

Q: Prove that ||A||1 = maxi<j<n{E#1 laij|}. i=1

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Q: Let f(r) = 4e and g(x) = In(5z): find (f o g)(2). %3D a) O 60 b) O 100 c) O 16 d) O 40 e) O 400 f)O…

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Q: is equal to where A = = and where B= In(r7s²r683) Alnr + Bln s

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Q: 11. Define F:Z × Z→Z × Z as follows: For every ordered pair (a, b) of integers, F(a, b) = (2a+1, 3Ь…

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Q: Suppose that A,B⊂S such that Pr[A|B]=25, Pr[A]=12 and Pr[B′]=12. What is Pr[B|A′]?

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Q: 6. Use the definition to show: if fn →f uniformly on S and 9n → g uniformly on S, then fn +9n → f +…

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Q: Suppose X is uniform on {6, 8, 10, 12, 14, 16}. What is Var[X]? A B C D E 27/12 33/12 35/12 35/3…

A: Solution-: Suppose X is uniform on {6,8,10,12,14,16} We want to find Var(X)=?

Q: 1. Let s = (001 001 001...) t = (01 01 01...) r = (10 10 10...). %3D Compute: a. d[s, t] b. d[t, r]…

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Q: (b) Consider the following, more general, true fact: For all n, m € Z\ {0} and g € N, if ged(n, m)…

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Q: 8. (p f(A) ≤ f(B), then ACB. Let f: XY. Prove that for all A, B C X, if f is one-to-one and

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Q: let n be a positive integer. prove that (X nY) nZ = X n(Y nZ) for all x,y,z E Z

A: You can prove the statement that (X∩Y)∩Z=X∩(Y∩Z) for all sets X, Y, and Z by using the distributive…