A := {x € (0,1)|x= ar, for some r € Q} re that sup(A) = 1.

Q: 1 Je* sin( t)dt 1+s* -st Prove that 2

A: We have to prove that:  ∫0∞ e-st sin(t) dt=11+s2

Q: az xdz + Y- = xy+z. a) If z = xy - xei , show that x- ду

A: Since you have posted multiple questions in this question which are not interlinked so I have solved…

Q: (a) Suppose X U (0,n), where n is a positive integer X-[X] U (0, 1). (b) Use the result of (a) to…

A: De Moivre's theorem gives an axiom for computing the powers of complex numbers. We first gain some…

Q: If X(t) with X (0) = 0 and µ = 0 is a Wiener process, show that Y(t) = oX %3D is also a Wiener…

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Q: H) 9f zyzec then prove thal nl x=4=Z, (alogen) %3D

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Q: 3. Recall that x - Poisson(0) = Mz(t) = e"(e*-1). %3D A. Derive the MGF of Z = (F - 0)/ /0/n where F…

A: Please find the solution below. Thank you.

Q: If the WSS process {X(t)} is given by X(t) = 100 cos(10t + 0), where 0 is uniformly distributed over…

A: Find the autocovariance function of X(t)? Is X(t) wide sense stationary? Explain If the WSS process…

Q: If AB = B for some B # 0, then A is inertible.

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Q: 6. Prove that for x > -1 then Vn E N, (1 + x)" >1+ nx .

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Q: 22. Solve 0 0, x > 0, Urr + Uyy = 0, u (x, 0) = f (x), u (x, 1) = 0, u (0, y) = 0, и (х, у). O…

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Q: Let Y(t) = (ao+a₁ X(t)) cos(2π ft+0), where a, are constants, is uniform on [0, 27], X(t) is a…

A: Solution:

Q: 9. Building upon Example 5.14, let X be a real-valued compound Poisson proce with Lévy measure v,…

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Q: In Exercises 23–30, find Tn centered at x = a for all n. 26. f (x) = e", a =-2

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Q: 15. Solve Orx + Pyy 0, -a < x < a, 0 < y < ∞, 8o, 0 < |x| < a, Oy (x, 0) = 0, || а. $ (x, y) → 0…

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Q: Find the Tians Farm F(s) = "suy (27>1>0) (1 – s)(1 – e **) 3) f(1) = e',

A: Given problem:- Find laplace transformation of f(t) =et (0&lt;t&lt;2π)

Q: - If W = xy + y2+1 azw Show that a2w ey %3D дхду дудх

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Q: Let p and g be continuous functions in an open interval I and ão E I. Show that if Yı and Y2 are any…

A: If y1, y2 are any two solutions of dydx+p(x)y=g(x),    y(x0)=y0. where p and g are two continuous…

Q: Consider the initial value problem *, where -00 < x < 0,t 2 0. If u(x,0) = Va - 1 and %3D u (x, 0) =…

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Q: Solve the following Cauchy-Euler D.E: x²y" – xy' +y = 2x2 + In x %3D

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Q: 1 prove that | e -ym dy = mr(m) -ут

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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: and let P(x) =E sin kx its partial sums. Show that k? Let f(x)= 5sin kx k=1 k=1 E,(f) <!. n

A: Given, fx=∑k=1∞sinkk2 and Pnx=∑k=1nsinkxk2 Let, fkx=sinkxk2 Since,fkx=sinkxk2≤1k2

Q: 1. Let g(x, y) = ev* /x. Evaluate gry and gyæ.

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Q: -2 Let f(t,y) = y + 3t?e'. Then f satisfies a Lipschitz condition in y with L = 2, given that 1<t<2.…

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Q: (a) Express the initial value problem: 1 s'(x) = -f(x), for x +1 f(0) = 1, as an integral equation…

A: The Initial value problem is given as, f'x=-1x+1fx, for  -13≤x≤12                         (1) with…

Q: Let f(x, y,2) = x² sin y+y e", where x =t +3,y = ne*-1 and z = In 2t. at t = 1 is: -2 1+2n O -12n…

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Q: K=Zp(t) ,then f(x)=x^p-t is irreducible over K[x]

A: GIVEN: K = Zp(t)

Q: 9. Verify that p(t) = te/50 satisfies the condition p(t) dt = 1

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Q: Consider h(x,y, z) = cos (xy) + eY² + In (xz) at the point P ( 1,0, = 1 Deter-

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Q: 5. Use the Fundamental Theorem of Calculus to show that the function u = u(t) is a solution to the…

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Q: 5. Use the Fundamental Theorem of Calculus to show that the function u = u(t) is a solution to the…

A: use fundamental theorem of calculus to show that the function u=ut is a solution to the initial…

Q: Let X(t) be a WSS Gaussian random process with ux(t) = : 1 and Rx(7) = 1 + 4 cos(7). Find

A: From the given information, Xt be a WSS Gaussian random process with μXt=1, RXτ=1+4cosτ. Consider,…

Q: 11. Show that the Beta function, defined by B(x,y) = (1-t"-1 dt , for x> 0, y > 0, satisfies the…

A: the given function is βx,y=∫01tx-11-ty-1dt for x&gt;0,y&gt;0 substitute t=1-s therefore, dt=-ds when…

Q: 4. Let z = x2 + y³, x = 2st, y = s - t2, s = u", and t = u sin w. Compute for Oz ду and дw ди

A: Let z=x2+y3, x=2st, y=s-t2, s=uw, and t=u sin w.To compute for ∂z∂w and ∂y∂u.

Q: 9 Give with an not example of T: W≤X →Y, to XT Ext. .R. boundedag. - keše bes

A: Please find the solution innext step this is an non-example of Tietze extension theorem in Topology.

Q: If y is a solution of y(x)-[(x-1t)»(?)dt = 1, then which of the following is true? (a) y is bounded…

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Q: Prove that the random process {X(t)} with constant mean is mean-ergodic, T T 1 if | C(t,, t2) dt dt,…

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Q: and further we require that u(r, z) is bounded as z - +oo. Find an expression for the steady state…

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Q: Let X and Y have the joint p. d. f . sx+y if 0<x, y < 1 elsewher f (x, y) = %3D The covariance…

A: Given X and Y have the joint p.d.f. fx,y=x+y  if  0&lt;x,y&lt;1    0          elsewhere

Q: Let f1(x, y) - e cos(5y), f2(x, y) = x² – 25 y². Find Vf, and Vf2, and show that %3D (fv) = 0 for…

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

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Q: 3) yv1+x²dx + x/1+y?dy = 0 (1+v1+x²)(1+/1+y²) Ans: V1+ x² +/1+ y² In +c (Bali, 2007) xy

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Q: Theorem 7-13. µ' = E {X (X – 1) ... (X-r+ 1)} = P (). ,where u is the rth factorial moment of X. s=1

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Q: Let M, = Z, – Z;-1, Zo = 0, where {Zn, n21} is a martingale. Show that E(M,„Z„-1] = 0

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Q: If J f(x) cos(mx)dx = and f(x) = En=1 cos(n²z) n3+1 then m = O 4 O 25 O 16 0 4 9 O 5 O2 03

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Q: B. Prove by direct proof general. P: Vn e ZE, (n – 1) (n+ 1) e Z,

A: I have solved B; you just need to expand (n-1)(n+1) and then use the fact that square is always…

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: If {X(t)} is a Poisson process, prove that n - r S P{X(s) = r/X(t) = n} = nC, where s<t %3D %3D

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Q: Show (in terms of e - 6) that a function f : R → R defined by f(r, y, 2) = (2x + 3y + 42) is…

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