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 k-s Vn! sn, Vn eN, n2 k.

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: 2. Using OLS or any other suitable method, Prove that E(ß1) = B1 E(ßo) = Bo

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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] .

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Q: Prove that the set S = {n² : n e N} is denumerable. %3D

A: We have to show that the given set S is a denumerable set.

Q: . Finish showing the Complementary Slackness Theorem by establishing that under the hypothesis of…

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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: is a conplex saries of nunbers and for VNEN monert an, bn ER Let 2n- an +ibn. Show that if Ź 2n =…

A: Given:  ∑n=1∞zn is a complex series of numbers and  ∀n∈ℕ, an , bn∈ℝ Let zn= an+i bn To show that :…

Q: Verify by mathematical induction 1. 2(2i– 1)° = n²(2n² – 1) for all n21. 33 n(n+1) 2. Prove: Vn e…

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Q: 4. Use mathematical induction to prove that Vn E Z* u {0}, []²=o(2i + 1)! > ((n + 1)!)**. n+1

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Q: Show that A is skew symmetric if and only if ai j = −aji for all i, j.

A: Suppose A be any square matrix of order n such that A=a11a12⋯⋯a1na21a22⋯⋯a2n⋯⋯⋯⋯⋯an1an2⋯⋯ann First,…

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: Prove that T2 is chaotic on 1.

A: Given that T2=2x0&lt;x&lt;122(1-x)12&lt;x&lt;1 The maximum for T is 1 at x=0.5. The dynamics of T2…

Q: 4. Use mathematical induction to prove that V n E Z* U {0}, [I²-o(2i + 1)! 2 ((n + 1)!)**. n+1

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

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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: Then, we assume that Wn+1 - H Wn - l kn+1 kin 1 2 E Wn-p Wn-p p=0 p=1 (6) n+1 - € Zn - € Vn+1 = 1 Vn…

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Q: Prove that ||A||1 = max1<j<n{£", |aij|}. %3D

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Q: 6. Given: 21 and 22 are complementary Prove: SX 1 WX S. 2,

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Q: az Find if yz – In z = x + y ax 3.

A: We have              yz-lnZ=x+y

Q: Prove that (aA + bB)t = aAt + bB t  for any A, B belongs to Mm× n(F) and any a, b belongs to F.

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Q: Prove that F(z) = 81 8z + 1 is chaotic on (-1,1).

A: To prove: Fx=8x4-8x2+1 is chaotic on -1,1. First plot graph of given function with -1,1 as follows:

Q: Let Xionne Xn be aris H c'cid) From PlO), or QcOi Findthe MP test g sige a to test Ho:G-Be N5 H,:…

A: Note: Hi there! Thank you for posting the question. As you have posted multiple questions, as per…

Q: Let v(s) = s²i + 2sj+9s¬²k. Evaluate - d -v(g(s)) at s = 4, as- ds suming that g(4) = 3 and g'(4) =…

A: To find the composite function, v(g(s)), substitute s with g(s) in the vector function,…

Q: @ Let A -a Find Cenditiens on a,b.c and d that AX =o has infailely many Solution. what must there…

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Q: Let Z ~ N(0,1). Prove for all z 2 0, P(Z > x) S

A: It is an important part of statistics. It is widely used.

Q: Let 9,n be antegers wth 2n and ged (a,n)=1. Show tht of U(Z.) ={a*+nZ:tez} then o#1 mod 6) for all…

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Q: Compute the Wronskian, W[y1, y2](t).

A: Let y1(t) and y2(t) be two differentiable functions in the independent variable t. Then the…

Q: Let gilx) i for n z2 i define gncr) -gn1(g,(), = 2ー× Deter mine the Intervai Such that g2avu CA) is…

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Q: S 1.Prove that L (coswt) = %3D s2+w2

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Q: Let X1, X2,, X, be iid N(0, o?). Consider the hypothesis test : ... Ho : o = 1 Hi: o = 4. Let C =…

A: Neyman-Pearson Lemma: Let X1, X2, X3,…,Xnbe a random sample from a probability distribution with…

Q: дz (B)If Z = f(x, y), x= s² - t², y = 2st. Prove that s მs 2 (s² + t²) of əx

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Q: . Let X be a r.V. such that E(X") is finite for all n = 1, 2,... Use the expansion

A: Introduction: Suppose X is a random variable. Then, the moment generating function (mgf) of X is…

Q: Prove that (A = B) where (A = ax + ay + az) and (B = bx + by + bz)?

A: Given- A=ax+ay+az and B=bx+by+bz To Prove- That A=B

Q: 3. Show that à = (6xy + z³)î + (3x² – z)ĵ + (3xz2 – y)k is irrotational. Find p such that A Vọ.

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Q: 1. Prove that L(t)=. s2

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Q: Determine Z- Trans furm in cludig vegiva of con vergence in n 7o メ(At n<o

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Q: 1. Obtain the weak formulation for the following BVP: in Ω, -V (kVu) = f u =g on ô0.

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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: Prove (x + y) + z = (x + z) + y without using induction.

A: It can be solve using group theory

Q: Then, we assume that Wn+1 - H Wn - u kin+1 kn 1 2 E Wn-p E Wn-p p=0 p=1 (6) Zn+1 - € Zn - E Vn+1 Vn…

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Q: 2) Suppose S,T E L(V) are such that ST = TS. Prove that null S is invariant under T.

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Q: Set n n and K intgers 1SKE ni Shorw that 2n42 2 - 2n %3D K-I Kニ)

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

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

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Q: Prove that S, pm (x)Pn(x)dx = 0 if m# n ? %3D

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Q: Prove that e“ p.w. on R. n! n=0 M8

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Q: 4: a) Let X.X., X -N (Hi,6f). Let Y,, Y2 Ym-N (u2,63). .... d (1-a) % CIfor oi when ai. a known.

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Q: Let X1, X2, - ,X, be iid N(0, o²). Consider the hypothesis test : Ho : a? = 1 H : o² = 4 Let C1 =…

A: Let X1,X2,...,Xn be iid N0,σ2. Consider the hypothesis test: H0:σ2=1H1:σ2=4 Let C1=s'2≥K1, C2=s2≥K2…