Show that the statement If n E N and a E Z*, then [Q(Va): = n is false.

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

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Q: Suppose, X|0~ Binom(m,0)and 0~ Beta( a,B), then E(X) is equals to, а) тав b) aß ma c) a+ B me d)

A: Given: Suppose,  Xθ~Binom(m,θ) and θ~Beta(α,β)

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: a Prove that < 16 – a| for each a, b e R. - V62 + 1 Va? + 1

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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: Given that en 2= V and en 3=W,express each of the fallowing in terms of vand w a) in zv0/3

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Q: If u (E) =0, then E is measurable.

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

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Q: Let x, y, z e Z Prove that if x – 2y + 3z is odd, then 4xy – 6z is even.

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Q: Let n e N, s > 0, a > 1. Show that n! 1"e" dr sn+1 Inz dr 1 (a – 1)2

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Q: Let X - N(H, o). Show that the MGF of X is etu+to²

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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: 6. If y = In(r+ 1) In z then what is dy/dr? %3D

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

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Q: If E, and E2 are measurable then (E,UE,) is not measurable .

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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: Let a, b, c E Z. Prove that if a b and alc, then Vk EZ we have a (b+ ck).

A: ab  &amp;  ac   Since ab, therefore b=a·m   ,  where m∈ℤ Since ac, therefore b=a·n   ,  where n∈ℤ

Q: ces. 1 W= d b such that Z = ax + bY. calars a and b such that w = ax + bY? = 0, is it true that a =…

A: We have to find the following conditions according to the given information

Q: locally Lipschitz in (t,x)E[a, b]x D.

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Q: Prove Va e Z, Vb e Z, 3|(a² + b² ) – 3|a ^ 3|b where xy means x divides y, i.e., there exists k e Z,…

A: In this question we use congruence modulo.The following  are useful in this question1) If  3 |…

Q: Give a natural deduction proof of V x B(x) from hypotheses 1. Vx (A(x) V B(x)) and 2. V y (- A(y))

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Q: too x(Ju) - xt) e ot * the famer heansfarm ナR X( Jw) Sen %3D

A: Fourier transformation of a function x(t) will be X(t) and inverse Fourier transformation of…

Q: Prove that Z(Sn) = {e} for n ≥ 3.

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Q: Suppose, X0- Binom(m,0)and 0~ Beta(a, B), then E(X) is equals to, a) maß b) aß ma c) d) me

A: Given: Suppose,  Xθ~Binom(m,θ) and θ~Beta(α,β)

Q: If e'z = /3 -i.Then Im (z) is %3D O a. e2 b. 0 С. 1 d. -In(2)

A: eiz=3-i Now let z=x+iy. Then, ⇒eix+iy=3-i⇒eix+i2y=3-i⇒eix-y=3-i…

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: (b) f(z) = tan¯"(5z) - (In(2"))®

A: We have to find derivative

Q: (17) Define S(n) as follows: n S(n) : i(i!) = (n+ 1)! – 1 i=1 Prove Vn e N1, S(n).

A: Consider the given expression. Sn=∑i=1nii!=n+1!-1 Now, use the mathematical indication to prove the…

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: When is S \ T = T \ S for sets S, T?

A: Given that two sets S and T. The definition of S\T=a∈S,a∉T and T\S=b∈T,b∉S. Here, neither of these…

Q: ala lo oepse a ad 1. Let m*(A) = 0. Prove that for every B, m*(BUA) = m"(B).

A: We have to prove the given statement.

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

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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: (c) If A E M,(R) is non singular and n 2 2, prove that det(adjA) = (det A)"- (d) If A E M,(R)…

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Q: ke an example at a Noe therian m adu le is not Artini an.( Prove)

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Q: Show that |Re(z)|+|Im(z)| < v2|z| for any z e C.

A: The solution is given as

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: Let m e N and a, b e Z be such that (a, m) = 1 and (b, m) %3D = 1. Let h = 0m(a), k = 0m (b), and…

A: Let m∈ℕ and a, b∈ℤ be such that a, m=1 and b, m=1. Let h=oma, k=omb,  and  n=omab. We need to show…

Q: %3D in Fin d E(S an d Var(s)

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

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Q: (6) If u, = x-Pa(x)Pn-1(x) dx, show that nu, + (n – 1)u,-1 = 2, | and hence evaluate u.

A: To get the desired result first we will the recurrence formula for pn(x). Then after simplification…

Q: Let x, y,p E Z, where p is prime. Prove that x2 – p²y – p + 0. (Hint: you may use without proof that…

A: Solution :-

Q: To test this null hypothesis, we compute t using a special case of the formula. state the formula?

A: In this case, we need to write the test statistic formula for testing the null hypothesis.

Q: Let w = r. In(xyz), then which of the following is 2 O a. O b.0 O d. O e.

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

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

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Q: Result n! de!... dn! Sn has permutations of ky pe [1i2* nte]

A: