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: i. If n is positive integer, Find (√3+ i)” + (√3 − i)”. ii. Separate tanh-¹(x + iy) into real and…

A: As per our guideline, we are supposed to solve only first question. Kindly repost other question as…

Q: Let X-(1,0,-1,-V2)'. Then, |X|l2= Select one: a. 2 b. 1 C. 3

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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…

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Q: If A₁, A, E and P(A) = P₁, P(A)= P₂, show that P(A₁4₂)≥ 1- P₁ - P₂. (Hint: Use DeMorgan's Law (44₂)…

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

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

Q: 5. (a) If r23 = 0, prove that R₁ (23) = (23) = 712 + 7³1 01-23 = (1 - 12 - 13). 07.

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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

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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: 2. Show that In (- - ) = ( + 2nn)i where n = | .... .... ..

A: Consider the following expression: x=ln-12-32iex=-12-32icosx+isinx=-12-32i

Q: 2. If A C B, then m*(A) < m*(B). 3. If A = Q' n (0,1), then A is measurable. %3D

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Q: Verify the &-ô identity: Ejkɛipq = djpôkq – djqðkp. Then use this identity to show that (a) EilmƐjlm…

A: The given identity is εijk εipq=δjp δkq-δjq δkp  where the εijk is called the alternating tensor or…

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

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Q: 0.2 Differentiate In x+1, with respect to x. x²-x+1

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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: Given that x>0 and y>0 proof that xy>0. (x,y are integers)

A: Given that x&gt;0 and y&gt;0, then to prove that xy&gt;0. x, y are integers.

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: all your calculations and simplify your answer completely. Il you only give a final an- Swer you…

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Q: if zi = ayi for i=1,2,...,n, show that z=ay

A: Here we have given that if    then we have to show that   so here zi and yi are random variable…

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: Prove that re" = 3 – x² for some negative value of x. [Hint: Use the IVT]

A: According to the given information it is required to prove that: xex=3−x2 for some negative value of…

Q: Let Q be an m xn nauix with ni >n and orthanormal columms, then we have both Q' Q = 1, and QQT -1 O…

A: The solution of the problem is given in the next step.

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

A: (.) We have to prove or disprove .

Q: A. Evaluate the follow 1. S dy 3+√y+3

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Q: 2 1] Assume A = 1 3 4. [2 2 4 and B = show that (AB)' = B' A' [3 6 4]

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Q: Let G = (a, b | a² = b4 = e, ab = b³a). a. Express a'b’abab³ in the form b'a', where 0 < is1 and…

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

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Q: 15. Suppose that a, b e Z and gcd(a, b) = 1. Prove that ged(a", b") = 1 for m, ne N. N and lot 8 frm…

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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: 4. Let 1-a a P = 0 < a, b < 1. 1-6 Prove (:) (1 – a – b)" 1 b a a -a a +bb a a +b -b b

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Q: Suppose x, y, z E R. Show that zz = yz if and only if either x=y or z = 0.

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Q: -3 < x To nenen Is fox is odd ? even or

A: Use definition of odd and even .

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…

Q: m Show that >( aij Aij i=1 j=1 i=1

A: Given that ∑i=1n ∑j=1maij=∑j=1m ∑i=1naij. Prove this as follows.