Suppose that > ak = a and let c E R. Prove, in all the that (c-an) = gory detail necessary, = c•Q. k=1 k=1

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: Consider the statement if (x = 1) and (y = 1) then z:= 1 where x and y can each be 0 or 1, with both…

A:

Q: Suppose X ∼ Binom(n, p). Show that E[X] = np.

A: Suppose Its probability mass function is defined as,Where,

Q: 86. Show that Z, = r(cose + isine) and Z2 = r[cos(-6) + isin(-6)] are conjugates. %3!

A: Given,

Q: S S-a₁ + S S-a₂ ++ n² ≥ S-an n-1' S ZZUA

A:

Q: 5. Suppose that A C B and that P(A) > 0 and P(B) < 1. Are A and B indepen- dent? Prove your answer.

A: Ans#-5; Given that A ⊂B, P(A) &gt; 0, and P(B) &lt;1              Are A and B independent?

Q: If S:= { / N - ¼/m : N,MEN, find wfs and sup 5

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Q: Prove by direct proof: “V n E Z if 2 ł (3n + 5), then 2 | n."

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

A: We are required to prove that 'Convergence in probability implies convergence in distribution' which…

Q: 23. Prove that any integers a,9(a?-3)

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Q: Let n 22. Prove that the equation y" = 2r" has no solution in positive integers.

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Q: 6.) Derike the binomial idendity nent) (un-) C Hint: For mzz, c^)+2(^) +m?]

A: To derive the binomial identity22+42+62+.........+2n2=nn+14n-16

Q: 17. Let P1.Pn be positive real numbers such that P1+ + Pn=1. (a) Prove the following: @P² ≥ (ii)…

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Q: 13. Prove or disprove: Q/ZQ.

A: (.) We have to prove or disprove .

Q: If S:= {1/n - 1/m²n,mEN }, find inf Sand sup S.

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Q: 4.9) Let p(k) = P(X = k) = c2 k for k = 1, 2, 3, and p(k) = 0 otherwise. Find the value of c that…

A: We have given that pmf p(k) = P(X = k) = c2 k for k = 1, 2, 3, and p(k) = 0 otherwise

Q: Consider the following statements: P:x is odd, Q: x is positive, R: -4 <x < 3, S: æ| < 2. Find all…

A: The given statement is a logical if-then statement. So this statement becomes false only when S is…