Let A = {r € Z:I mod 7 = 2} and let B = {r € Z: 7|(x – 2)}. Show that A = B. %3D

Q: 12. Define def M e-$dWg• s=0 For each t > 0, compute (a) E[M¡]. (b) E[M?]

A: Given  Mt=∫s=0te-s dWs      , where W is a Brownian motion.

Q: Let S, := (1 - (-1)"/n :n e N). Find inf S, and sup S,.

A:

Q: Let S = {:n e N}. a) Show that inf S exists Question 16 %3D b) Show that inf S = 0

A: PART A The given set is : S=1n:n∈ℕ As we know that : ℕ=1,2,3,4,......... The set is given by :…

Q: Given: LM = NO; LLMO = ZNOM Prove: ΔLΜΟ-ΔΝΟM M. N.

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Q: Let X - N(H, o?). Show that the MGF of X is etu+o²

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Q: 6. Show that, for integer s, S. s! | x25+lexp(-ax²)dx =: 2a +1

A: Note: Given proof statement has a little mistake, I have proof the actual result. First i have…

Q: For each n e N, let A, = (}, } + ) %3D Find Unen An. Find Nnen An.

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Q: 0 <m < n such that am = e. (Hin

A: Subgroup generated by any element in a finite group will have at most n element where n is order of…

Q: If A= then A" + nl is equal to 1.

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Q: If f(z, y) – e . In(9y) , then fz is given by: O - 9e In(9y) o - 27 e In(9y) O 9 e In(9y) O 27 e…

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Q: Construct the fundamertal soluton G(x, E) -?G= S(x- E) X, E ER? with, as x -E00 or

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Q: Prove for all mEw and n E w, we have that m+ ·n=m・n+n.

A: Given that m,n∈ω. To prove: m+· n=m·n+n

Q: Given: e || m Z3 = 24 1 Prove: e || n m 3 4

A: Given that, Here t is the transversal and∠3≅∠4

Q: .3 T Evaluate In In e²+j²4

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Q: O let X be fe VrV Finel ECX) ael E(x)

A: Note: As per guidelines we can answer only one question at a time. I request you to please repost…

Q: 2. Let P: C(0, 1]C(0, 1] be given by Pp "- e. Compute dP(x). Be sure to justify your calculations.

A: Given: The given function is  P:C20,1→0,1 given by pφ=φ''-eφ

Q: Prove that E, k2* = (n – 1)2"+1 +2 for all n > 1. =D1

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Q: Let m eN and a, b e Z be such that (a, m) = 1 and (b, m) Let h = 0m(a), k = 0m(b), and let n = 0m…

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Q: Let S = {:n e N}. %3D a) Show that inf S exists b) Show that inf $ = 0

A: given S=1n:n∈ℕ to show  (a) infS exists (b) infS=0

Q: Let X and Y be independent and distributed as N(u, 1). Is T = X-Y sufficient?

A: According to factorization theorem, If the likelihood function L(x,θ) can be expressed in the form…

Q: Show that E [(X-µ)^n]=0 for ∀n, with X~N(µ, σ²) and n≥3 being odd numbers.

A: We need to show; E[(X-μ)n]=0 for ∀n and n≥3 being odd numberswhere;X~Normal(μ,σ2) That is, we need…

Q: Suppose T: R + P, has an action on a

A: Given, And

Q: use an appropriate method to And the genenal Lolution of the ff. difterential equan a:) axy dx F…

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Q: Σ (X) E(Y), E(XY ) , Coν ( X, Y) , σχ, σχ and ρxΥ.

A: As per the Bartleby guildlines we have to solve first three subparts and rest can be reposted...…

Q: 11. Suppose A\BC C. Prove that if x E A\C then x E B.

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Q: Let s in G, s is not equal to 1.

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Q: Prove that for each n E N, (d) (n) e (d) dn

A: To prove, for each n∈ℕ, nφn=∑d|nμ2dφd. We know that the mobius function μn is defined as: μn=1if…

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: Show that for all n E Z, n² 5.

A: Consider the given condition: This shows that whatever will be the values of n, it will always be…

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

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

Q: Determine if the following is a fundemental set of solutions to  y'' -6y' + 25 = 0 Given that…

A: If the Wronskian of two solutions is not equal to zero, then the given set of solution is…

Q: 2. Let X~N(u, o2). Show that the MGF of X is etHtc²a².

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Q: Given x (0) -2,x (1) =1, x (2) =2 evaluate X(k) by using DFT.

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Q: For any k = 1, 2, .... n, find E T₁+T2+ T₁+T2+ ... + Tk + In,

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Q: 2. Let P:C0, 1] → C|0, 1] be given by Pp = " – e°. Compute dP(y). Be sure to justify your…

A: We have Pφ=φ''-eφ We need to find dPφ If we have  y=fx⇒dydx=f'x⇒dy=f'x dx Similarly, we find dPφ for…

Q: If m` E = 0, then E is measurable.

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: d. What happens if P < M for all t?

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Q: es lls ew zeiflucifiib ons einiseen stigesber Show that for neN, (2n)! ' (r!)² ((n −r)!)² - 7=0 2n -…

A: We will use binomial expansion to prove given relation

Q: (12) If E(X*) is finite with E(X) = µ and Var(X) =o² then prove that E((X – µ)*) > of. %3D

A: Note : Since you have asked multiple questions, we will solve the first question for you. If you…

Q: Exercise 9. Use the above formula, together with definition of T, to show that Va(t)||°||a"(t)||² –…

A: Curvature:  If T(t) is the unit tangent vector function to the curve α(t) then the curvature, κ, is…

Q: Let fELoi Show that i for any E>o, the set 3- S has a positive measure -

A: For f∈L∞ , f∞=infM&gt;0:μx:fx&gt;M=0For M&gt;0, defining the set AfM=X:1 fx&gt;M⇒For f∈L∞,…

Q: If det A= 0 why is det A+ = O? If A has rank r, why does A+ have rank r?

A: Given Data det A=0   Suppose the column and row of the matrix are linearly independent. The matrix…

Q: Find an expression of for y = 2*ev*, Hint: (a*) = a* . (x) · In a

A: We can find the derivative as below

Q: x | X> 3) =? and E [X | X> 3] =?

A: X ∼ exp (λ)  having pdf  f(x)=λe-λx      x&lt;0

Q: 2. Give the condition which ensure that |ez| < 1 where z in C.

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Q: Suppose / (s) - (÷)*" valuate: Js(x) dr +C 3- In (5)

A: Given      f(x) = 1512-3x

Q: If n = a? for some a E Z then 1155 n" + 7777n + 1 %3D

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Q: e){x*,x%,x}on(0,)

A: