4 Let X,) be a pankizlly oadered e For x,y € X deftre xLy eff x sy and x#y. Pucose the Fallowiag: xLy and y(z = x (2 for x,Y, a € X.

Q: Let w = xydx + 2xdy - ydz E 2'(R) and let a: R a(u, v) = (uv, u², 3u + v). Compute do, a*o, d(a*o),…

A: Given ω=xy dx+ 2x dy - y dz in Ω1ℝ3x,y,z And let α:ℝu,v2→ℝx,y,z3 be given by  αu,v=uv, u2, 3u+v. To…

Q: 2. Golve the pollowing uaing methods for non- exad ea vations, Jintar eouaton or Bernonlli's es…

A:

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: We defined NCE) = ZE for ZE Z Ii]. Prove N (Zw) = NLZ) Ncw) for all Z,W E Z6] .

A:

Q: 5. Show that if x,y∈Z and 3|x^2+y^2 then          3|x or 3|y 6. Show that if c=sa+tb where c=(a,b),…

A: Please check that x=1,y=3, then x^2+y^2=10 does not divide by 3. So  must 3|x and 3|y instead of 3|x…

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: Let E={(r,0,z) : os es T, OS zs 1, 5Vzs rs 5. Compute

A: Introduction: The triple integral ∫∫∫Ef(x,y,z)dV with the region E=(x,y,z):a≤x≤b, c≤y≤d, e≤z≤f can…

Q: Let Y - Wei(2,2) Compute V(Y). O 3.1416 O .8584 (answer) O -2.1416 O 4.1416 Let Y ~ W ei(2, 2)…

A: The variance is calculated as follows:

Q: O Let X,X. X3 beasamPle of size n=3From the Bernolli Pmf Assune Ti ヒ(X、メ,メ) X、+メz+Xg and こ I= X,X t…

A: Using the Neymann factorisation theorem we can find out the sufficient statistic.

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…

A:

Q: 4. If X and Y are independent RVs, then O2+ ,(@) = ¢,(@) × ¢,(@)

A:

Q: 13. Let T, O E L(V, W). Show that rk(T+0) ≤rk(T) +rk(o)

A: We are given that τ,σ∈L(V,W) We suppose, w∈Im(τ+σ)   ....(1) So, there exists some v∈V such that…

Q: © If g(x, y) = In then xgx - vgw in simplified form is [c].

A: The objective is to solve partial derivative problem and explain process.

Q: Is S = {a + b√(2) : a, b ∈ ℤ} dense in ℝ?

A:

Q: 4) Which of the following foumulas does not Kold for all x, we CZ O la wl= |2). Iw). (6) X W = ņ. W.…

A: We know that

Q: Let x, y E Z. Prove that if 3 ł x and 3 ł y, then 3|(2² – v²).

A:

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

A:

Q: 10. Given: YX = ZX WX bisects ZYXZ Prove: AWYX AWZX

A:

Q: 5. Prove that if x, y ER, then x + y ≤ x + y. Use the following cases: (i.) x, y ≥ 0, (ii.) x, y <…

A: We now prove that if x,y ∈ ℝ , then x+ y≤ x+y with the help of given three cases.

Q: 2) Given: ZRAL E LYAL, ZRLA LYLA A Prove: AR = AY R.

A: Explained below

Q: Let x, y € R" be given. (a) If ||z||||y||=1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1 and ||y||…

A:

Q: 3. Show that S, has the presentation (x, ylx* = y? (xy) %3D %3D

A: Given: Group S4 We need to show that S4 has representationx,y|x4=y2=xy3=1

Q: The joint propabiiy fnelion sTrtwo disen Ete iandom veritbles ☆ and y is given bi flkr)= c(2K Hy)…

A: The given joint probability function is as follows:

Q: In R* whire n eN, let こ and イ.ES":(tx € E),(ヨr >o)ud tlnt Br (x) CE lang the deginition 4 topelogs…

A:

Q: If U(x,y)=In(x²-y³) then U+Uyy= ....... ....

A: The solution is given as

Q: Let x, y, z E Z be such that both odd, and z is even. Prov

A: Introduction: Even and odd numbers are distinguished by the following characteristics: Even Number:…

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

A:

Q: c If in is an involution of X, show that (x, y)in = (yin, xin) is an involution of X2). %3D

A: Let X be a lattice We have to show that (x, y)in=(yin, xin) is an X[2] X[2] is a 2 x 2 matrix We…

Q: 2. Given: XW = XY, HA I WY, KB 1 WY Prove: AHWA~AKYB. W.

A: Given that XW¯=XY¯ in ∆XWY. So, ∠XWY≅∠XYWor, ∠HWA≅∠KYBGiven that HA¯⊥WY¯, KB¯⊥WY¯So, ∠HAW≅∠KBY=90°

Q: 3. Show that X = (x|Ax=b, x ≥ 0} is convex. following ure which no

A: As per our guidelines we are supposed to answer only one question.

Q: If y = Vln(x)+ In(/E), then y' Select one: 1 O a. (z)u1^z O b. (z)u1^z O c. 2z /In(z) Od. 2z /In(z)…

A: Differentiation

Q: 13- dy + 2xy = y diferansiyel denkleminin genel çözümü aşağıdakilerden hangisidir? -x* +c1 y = e a)…

A:

Q: show that if Z ~ N(0, 1) and Y ~x²(n) with Z and Y to be independent, then T: n

A:

Q: 2 Given u -1 and U , for which w is the set {u, v, w} linearly independent?

A:

Q: 1. Let z, y € R" be given. (a) If ||z|| = ||y|| = 1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1…

A:

Q: 4) Let (an) be anumsenical seguence sveh that lin u"4, Then the nadius of Comvergonce of the beries…

A: The given series is: ∑n=0∞7an2xn. We will use the root test to find the radius of convergence.

Q: prove in words: If {X, Y, Z} ` A ∧ ¬A, then {Y, Z} ` ¬X ` = derivability

A: Use the deduction theorem: Deduction theorem states that, If ┌ A⊦B then ┌⊦ A→B

Q: 4. A region D in RN is said to be convex if and only if for every u, v E D and t e (0, 1), then tu +…

A: Given: A region D in ℝN is said to be convex if and only if for every u,v∈D and t∈0,1, tu+1-tv∈D.…

Q: For x = 1, 2, 3 and y = 1, 2, let the joint pmf of X and Y be defined by fX,Y (x, y) = x + y/21…

A: Solution

Q: (X), E(Y ) EXY), Cov(X, Y) , σχ, σχ and Py.

A:

Q: 1. Let 1, y € R" be given. (a) If ||z|| = ||y|| = 1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1…

A: \documentclass{article}\usepackage{amsmath,amssymb}\begin{document} (a) Norm x=Norm y=1\\ Let angle…

Q: Let X, , X>, . X, be a vandam of size I For o hormol distribuien with unrur M and $6. IS 5 Xi = 132…

A: Provided information is ;  sample size(n) = 11 ∑i=1nxi  = 132 ∑i=1n(xi-x¯)2 =99 confidence interval…

Q: ZVYW E ZTXZ, WX YZ, and ZZ ZW. Complete the proof that AVWY ATZX. W

A: Given: ∠VYW≅∠TXZ , ∠Z≅∠W Also, WX≅YZ   1 Adding XY to both sides of (1). We can do this using…

Q: Let X, Y, Z C U. If Pr(X) 0.19, Pr(Y) 0.33, Pr(Z) 0.45, Pr(Xn Y) 0.07, Pr(Xn Z) Pr(Y n Z) 0.16, and…

A: Given DataP(X) = 0.19P(Y) = 0.33P(Z) = 0.45

Q: Let x, y, z ∈  ℕ. Suppose gcd(x, y) = 1. Prove that if x | yz, then x | z

A: In the given question we have to prove that if x|yz , then x|z and g.c.d(x,y)=1 and x,y,z∈ℕ. we will…

Q: A subset W of R° is given by = {(x,y, z) | 2x = y = 2z}. (i) Write down the distinct elements of W.

A:

Q: 6. Given: ZR= ZZ CA= YA Prove: ACRA =AYZA Y

A:

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

A:

Q: If f(x,y,z)=z In(x²ycosz), then fz is O 2z X OZ y O2z y O In(x²ycosz) - ztanz <|N

A: This question is related to partial differentiation, We will solve it using given information.

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

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.