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

Q: For each n ≥ 1, take An= (-1/n, 1/n). Then, 1 An {0} and U1 An (-1, 1). Prove these.

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Q: IRI- 2. <| sinz|< -e 2

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Q: let ζ ∈ N (10.2) calculate P (ζ ≥ 7.5) and determine E (2 ζ2 - 200)

A: #Given: z follow normal distribution with mean mu=10 and standard deviations=2  1): to find…

Q: Define the function fn : [0, 1] –→R for n > 1, defined as if æ E (,1] nQ if x E (, 1]\Q if x E [0,…

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Q: Let G = D8 = (r, s | r8 = id = s², sr = r²s). : (a) (b) (c) Let N = (4). Show that N ≤G. Find the…

A: Solution:-  (a) To show that N is a normal subgroup of G, we need to verify that for any n in N and…

Q: 3. Let f a nonnegative function defined on R. Suppose that for all n 21 you have to (r)dr <1 Prove…

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Q: G finta topoegy let x ba non-empty sel finit!. %3D A is chseel in Tos iff A is finit are closeel).…

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Q: #1 same holds for all x, y E R. (First prove that a" ay at+y for all x,y E Q, prove that the at+y…

A: We know, The supremum of a subset of a partially ordered set is the least element in that is greater…

Q: Prove 8xyz < (x + y)(y + z)(x + z) for every x, y, z E [0, +∞[.

A: We have to prove given identity: Concept:

Q: A function f : N × N → N is defined by f (m, n) = 2ᵐ−¹(2n − 1). (a) Prove that f is one to one and…

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Q: Suppose you are given four natural numbers w, x, y,z e N such that w>x and y > z. Show that ly – w|…

A: It is given that w, x, y, z are natural numbers such that w&gt;x and y&gt;z. Now we want to show…

Q: 2.J Let fn(x) = x ½n for 0<x< 1 and neN. or.) Find a function f on [at] such that (fr) converges.…

A: If each ,  is continuous and pointwise convergent on D and the sum function f of the sequence is…

Q: Find de X ->/ X- / X -/

A: The limits can be calculated by inserting the limit in the function. If the solution is in the…

Q: Prove = cos for some x in (0, π/2).

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Q: Prove that %3D 3 and hence. deduce that. 3--(n+3)Jn43+ (n+s)Jnes to 2.

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Q: -пя for all x € 0, 00). t fal(x) — же па for all z € [0, co). Find || fn||sup for all n. Prove that…

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Q: Let filN >N be defined by f(n) = Ź v(d) for all nEl din where r(R) is the number of the positive…

A: A multiplicative function on the set of natural numbers is a function T such that Tnm=T(n)T(m)…

Q: 8 Express the limit_lim Σ (ck-4ck) Axk, where P is a partition of [ – 6,9], as a definite integral.…

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Q: The function f: {0; 1}³ → {0; 1}defined as follows: f(x,; x,;X,)=(3 x,+ x +2 x;) mod 2, then f(1; 1;…

A: We are given a function  f: {0;1}3 → {0:1} defined as:f(x1, x2, x3) = (3x1+x2+2x3) mod 2To find:…

Q: 4.1 Suppose that fig and g'are continuous on the interval 1= {x :asxs b). Show that a ) exists, and…

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Q: 2.61 Let fn(x) = n 1. Prove: fn 0 uniformly on R.

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Q: Find all the strictly monotonic functions f: R→R such that f(x + f(y)) = f(x) + y, for all z, y € R.…

A: a. Consider the given condition, fx+fy=fx+y, for all x,y∈ℝ. Here we need to find the strictly…

Q: Let an m be non-negative numbers for all integers n, m > 0. Deduce Ši an,m = ΣΣ An,m n=0 m=0 m=0 n=0…

A: according to fubini's theorem if am,nm=1,n=1∞ is a doubly indexed sequence of real number then we…

Q: Suppose that 0< fn < In on X for all n e N. If gn 30 on X, then fn 30 on X.

A: Solution:-

Q: Let n >= 3 be an integer. Let Vn(x) = {[x 1 1 1 ... 1 1]T,[1 x 1 1 ... 1 1]T, [1 1 x 1 ... 1 1]T,…

A: By the consequences of vector spaces we solve the given problem.

Q: 2. Let (r) be the function defined by V(x) = 1. Let 1+x. In(x) Vovo...(x) n-copies Note that…

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Q: be a sequence of F-integrable functions on [a, t ow f is F-integrable and

A: Let  be a sequence of integrable functions whose choose  are arbitrary.Since  is uniformly…

Q: If f and g are in E(0, ∞), then L[ f g] = L[ f ]∗ L[g]

A: Let f and g are in E0, ∞. We have to show that L[f g]=L[f]*L[g].

Q: x[n] y[n] Σ X) ay[n – 1] D yln-11

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Q: Prove using the IVT. For all integers n, sinnx = cos x for some x ϵ [0, π].

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Q: Consider Z-N(0,1). Which of the following statements are true? Select all that apply. For z > 0, P(Z…

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Q: 4x + 4 f(x) = {√ √²/√x + √// 0<x< 7/7/20 X 4 0 otherwise x < <0

A: To find the quantile and cumulative distribution function (CDF) for the given probability density…

Q: Let n EZ. Show that f(x)= x³ – nx +2 is irreducible over Q for n+-1,3,5. 72

A: Given: fx=x3-nx+2, n∈ℤ. To show fx is irreducible over ℚ for n≠-1,3,5.

Q: 4. Prove that if (an) is increasing and an ̸= 0 for all n ∈ N then ? 1/ an is decreasing.

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