ercise 5.1 Show that if Q-x, then E[QI = r and var [Q] = 2r. Hint: Use the representation Q = E", Z? with Z independent N (0, 1)

Q: Given 21 = Z3 Prove jk 13 2.

A: Given,  ∠1≅∠3 To prove  j∥k

Q: Q.1 a- Verify Euler's theorem U= x"In(?)

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Q: Consider the Venn diagram below. Illustrate the other De Morgan's law by selecting the region(s)…

A: GivenAc ∪ Bc

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

A: We will give two proofs one based on partial sums and the other will be a direct proof.

Q: 5. Use Taylor's theorem to write down an explicit formula for R, (1). Conclude that R, (1) + 0, and…

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Q: Prove that the following relationship is true: nCr + nCr + 1 = n + 1Cr + 1

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Q: Q3- Give an example of two denes not equal X subsets of (R,T) and (z, of ). ( Explain your answer ).…

A: Answere : The two example are given below :

Q: Let n > 1 be a positive integer and let a1, a2, ... , a, be real numbers. a) Prove that Σ α la1| -…

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Q: By using caushy Rieman prove that ·C. Prove thatd az az e : ae

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Q: Let (W, t > 0) be standard BM. Show that P(n|w,1 – Wa sc) → as n → 0.

A: From the given information,  (Wt, t≥0) be standard Brownian Motion. Note: In the given situation,…

Q: I independent, and that o+zz-5 = 7. Based on the provided information, answe owing questions: ind…

A: Since the question has multiple sub parts we will solve the first three sub parts. Please resend the…

Q: Prove that gcd(m + 1, n + 1)|mn - 1 for all m, n E Z \ {-1}.

A: Let, gcd(m+1, n+1)=x From the properties of GCD, we can say that, x divides m+1 and n+1 Now,…

Q: If E and E, are measurable , then (E,UE,) is not measurable. ylgn

A: False.

Q: Show (x – a,)²(x – az)² .. (x – an)? + 1| is irreducible in Z[x] and Q[x]

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Q: 1. Let, a1 = 7 and for n 2, an An-1 express an in terms ofn. %3D ". **

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Q: For allne z, gcdin,nt)=l.

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Q: In the proof of Theorem 2 we determined the quan-tity E[X(X − 1)], called the second factorial…

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Q: Let E Os 0s n, 0s zs 1, 8Vzs rS 8. Compute

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Q: Assume that |Xn] 0. (Hint: first, prove that P(|X|>c+1) = 0.)

A: Answer:

Q: Prove that, IXI<a Rf and If -aL X Z a

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Q: 1. Condense and expand. a) Condense as a single quantity: 1 In(3)+- In(x? +1)-3 In(1+x) 3 b) Expand:…

A: Given, (a)  To condense ln3 + 13ln(x2+1) -3ln(1+x) (b) To expand lnx2+1x+2(1+x)5

Q: eb Let A E M₁ (TR) be given If AAt = On HOME JATOT prove that A = On - Does this hold if A EMn (C)?

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Q: Prove that for all values of the constant A, eAInt = e In.

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Q: (2) Show that if dim(V) = k (so in particular, T has k distinct cigenvalues), then there exists v e…

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Q: Find all values of z such that cosz = 4.

A: We find all values of z such that cos(z)=4

Q: nd C such that AC=E

A:

Q: = Prove that E = ehd.

A: We know that Ef(x)=f(x+h)

Q: Prove the following in Z[i]: if a y and 3|y with a and B relatively prime then aß | y. in

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Q: ), P{|Zn – c| > ɛ} → 0 as n → o. Show that for any b on g, E[g(Z„)] → g(c) as n→o

A: *Answer:

Q: (a, o) is not measurable. ylgn ihi

A: Given :-(a, ∞) is not measurable.

Q: Let H(x):=k for x=, (keN) and H(x):=0 elsewhere on [0,1] Show that H(x) 4ER[0,1].

A: In this question, we use the definition of Riemann integral to check the function H(x) is Riemann…

Q: Use the diagram to prove that ACAS ATAS. A T. Given: CS = ST and mzCAS mzTAS= 90 Prove: ACAS= ATAS

A: Given : CS ≅ STm∠CAS = m∠TAS = 90°

Q: Calculate Ehe Value of:(o00001 (0000001A in GF () witlh vmc) =x°+x+X++

A: It is given that

Q: Let x E 10, 1[. Prove the following identity: ["in (1²-1)dt = n(n+1) m a xn+1 1

A:

Q: Given: mAB = x Prove: mz1 = 180° - x A D Exercises 25, 26

A: Given:           m AB⏜ = x   To prove:              m&lt;1 = 1800 - x

Q: Determine all the number(s) c which satisfy the conclusion of Rolles Theorem for

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Q: 3. ncnt) use PMI to prove that E

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Q: Given the function z = In (2xy + e3x) - 5y, %3D az a. ax

A: Since, we have to find the partial derivatives ∂z∂x and ∂z∂y of functiongiven by z=ln…

Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

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Q: When A has full rank, the pseudo-inver en m≥n, and as A+ = AT (AAT)-¹ when n A = I, so A+ is a left…

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Q: Use Euler-Maclaurin's formula to prove that E n² = "(n+ 1)(n +2) 6. %3D 1

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Q: IRI- 2. <| sinz|< -e 2

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Q: Show that if E,=1 a, diverges, then En=1 | |a,| diverges. %3D

A: Given, We want to prove that

Q: Question 2: Show that the set S = {rVT + V2:rE Q) is dense in IR. Explain and show your work.

A:

Q: Prove that : a"+ (a+ 1)° =1 +a(1+い) (mod a la+s) for all a, m,n E Z.

A:

Q: In Z, let H = &lt;5&gt; and K = &lt;7&gt;. Prove that Z = HK. Does Z = H X K?

A: Given: In Z H=5K=7

Q: ***PLEASE HELP WITH THE FOLLOWING AND PLEASE PROVIDE HANDWRITTEN WORK ON PAPER*** (4) Find the…

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Q: If E and E, are measurable , then (E,UE) is not measurable

A: False.

Q: 7) Show that K± = det [ T 4 J.

A: Sol

Q: 4) Let x, y, z E Zg. Prove or disprove the following: Ifxy = xz, then y = z.

A: