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Q: 1. Let U2 = 2. fui, 4.} is an orthonmal basa fo IR Show that

A: The set of vectors v1,v2,…,vn of a vector space V are said to be an orthonormal set of vectors if…

Q: Let X N(u, 1) of size n. Obtain UMVUE for u.

A: Given that  Let X  N(μ,1) of size n. Obtain UMVUE for μ

Q: Let X N (u,1) of size n. Obtain UMVUE for M.

A: We will use Lehmann-Scheffe theorem to find the uniformly minimum variance unbiased estimator for μ.

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

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Q: 1. Show that S, has the presentation (x, y|x² = y = (xy)² = 1). %3D

A: The symmetric group S3 is given as S3=I, 1 2, 1 3, 2 3, 1 2 3 1 3 2 where  I=123123, 1 2=123213, 1…

Q: Prove that r(z) = (In-)²-'dt , Re(z) > 0.

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Q: If Z= cos20+isin 20 prove that = sec² 0 %3D (Z +1)

A: We have to prove L.H.S = R.H.S

Q: Prove by mathematical induction: X n (U-, Y) = U=1(X nY), for V n E Z* – {1} and sets X,Y,,Y2,. Yp-…

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Q: Let X, and X, independence r.v. have same moment generating function (1-) where isa, let Y=aX₁…

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Q: Q1. Verify Euler's theorem for the homogeneous funetion, fxy.2)-axy +byz + czu

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Q: A hransfoomation is definad by Hhe axa matox A=-a b-a atb Ifois an odd imtreger, prove %3D ushere a…

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

Q: Functional theory and functional analysis

A: According to the given information it is required to prove that:

Q: Exercise 4 Show that for cach k EN it holds that kn-k < n! S n", Vn E N, n2 k. Deduce from this that…

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Q: 1 Show that IS the separable me tre space (R, d), where el(x, y) = be-y/

A: Here we use basic real analysis

Q: if JERLA.b] and it hex)- J(x). except for finite mumber of Paint in Cab] then that he Rlab] prove.…

A: if g ∈R[a,b] and if h(x) = g(x) except for a finite number of points in [a,b] then prove that h…

Q: y Verify Euler's theorem for u = x' In

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Q: (b) Compute r if: arctan r + arctan 1 = 2 ( arctan z arctan

A: tan-1x+tan-11=2tan-1x-tan-113

Q: 1) Prove that Ym,n EN such thert m¢ nt = n<m using the princdple of in duction!

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Q: Show that there exists a set N C (0, 1) that is not Lebesgue measurable.

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Q: The Wronskian of y,(t)=e 2, y,()=e" is-e St. Select one: True False

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Q: (o, 1,) 1. find th directional denivatve in the dire chons of at 2 Gve inter pretahon of amp uted…

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Q: Show that if A is nonsingular and AB = O for an n × nmatrix B , then B = O .

A: We have given  matrix A is non singular and AB=0 for an n×n matrix  B. We have to prove that B=O

Q: Let A: C [0,1]→C [0,1]. Prove that operator continuous reversible, find A. 2.11. Ax(t)= x(1)- -…

A: The given operator A such that A:C0,1→C0,1. We use the inverse operator method for solving linear…

Q: Show that if AT Ay = y for all y in Rn , then ATA = In .

A: Given that     ATAy=y  ∀y∈Rn. To prove ATA=In

Q: In R', the point (1, 1, 1) does not belong to the cylinder x + y = 2. Select one: True O False

A: Given problem:- In R³, the point (1, 1, 1) does not belong to the cylinder x² + y² = 2.  True  False

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

A: According to factorization theorem, if the likelihood function Lx,θ can be expressed in the form.…

Q: Assume that u = U (x,y) iu differentiible and x =r ca Ô and y=rsin o eapress 2. entinely in termv 0…

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Q: Showw Fom yex)=D90Y,(x) + a, y,(x) that seris sol utron is of the hene y,cx) contains only even…

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Q: 2. What is the largest possible c in (1) if X=R² and x₁=(1,0), x₂ = (0, 1)? If X = R³ and x₁ = (1,…

A: Given: c To find: a) X=R2b) X=R3

Q: 1. If x, y, z E R, prove that (a) \x+y\ < \x| + \yl- (b) |x– y| < |x – z|+|z– y|, and (c) ||지| -…

A: Since you have posted a question with multiple sub-parts, we will solve the first three subparts for…

Q: If (x* – y*,x – z°) = 0,z = z(x, y), prove: az y3 az +x³z4 ax - %3D ду 3

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Q: Consider Uu=4uzzy u(x,0) = u(x), u₁(x,0) = u₁(x), on the domain -∞<x<∞, 0≤t<∞o. Suppose that both…

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Q: Let C: |2 | 4 and I 6 de Then O None of these OF0 (WithoUi applying cauchy Goursat Theorem) O Ho by…

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Q: prove that Sa/Gn has f discription with M-ly ny-l Degrees tof freedom

A: Suppose that xi and yi are two random samples of size n, drawn from a normal population.  Population…

Q: In general, for a a) S is compact 2) Sis closed at S is compact

A: We use definition of compact metric space, totally bounded set, Closed set, accumulation point or…

Q: /show that fo-1x is-Contindals everywhere.

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Q: find the desivalive d the functian ha)= ainfse and compute h'(2) Th o

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Q: Let an 1+1/V2+1/V3+...+1//n , Prove {an} is unbounded. %3D

A: Given that, an=1+12+13+...+1n

Q: Suppose f" exists and is bounded on Co,0), Prove that if bam fa)=0, then bim fGx)=0.

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Q: By using Cauchy Residle Theorem 到一2p 121=1

A: First we have to find residue at singular points.

Q: Assume that a1, a2, ..., an E R (with n > 2) such that a1+a2++an = 1. Prove that af+az++a > 1 (Use…

A: Cauchy-Swartz Lemma : For any real numbers  and  the following inequality holds  ,

Q: Select all the Points through which a unique solution for the DE ,2 y? IS nySot eed by the Existence…

A: Given differential equation is y'=13-x2-y2

Q: Let S = {(x, y) :x² + (y – 1)² < 1} U {(3,0)}. %3D Is the origin an extreme point of conv S? Is the…

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Q: Show that for all x, y E F, there exists e with e < Emachine such that fl(x * y) = (x * y)(1+ €) is…

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Q: Show that Z/mnZ is isomphoric to (Z/mZ)×(Z/nZ) where gcd(m,n) = 1

A: Concept Used:( The First Isomorphism Theorem) If ϕ be the  ring homomorphism from G to G then…

Q: A mapping f:C→ C is called bounded, if sup |f(2)| < o0. zEC Show that sin, cos : C →C are unbounded…

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Q: Given: ZSUY = ZNUY ZSYU = ZNYU Y Prove: ASUY ANUY

A: Given that         ∠SUY  ≅ ∠NUY        ∠SYU ≅∠NYU Here we have to show that               ΔSUY ≅ΔNUY

Q: a- if w = f(u, v), u = x + y, v = x-y Prove that = ( - )* aw aw %3D ду du.

A: