Suppose we observe X1, X2, ..., X U (0, 1). Show that nX(1) 4 Exp(1). с.

Q: 10 of 17 If y(t)+(t-r)y(r)dr=t, then y(t)= Su-nyndr3Dt,

A: Convolution Theorem: The Laplace Transform of the convolution of f(t) and g(t) is:…

Q: Show that for some number a > 1, we have 1- <= In(a) <= a – 1. Hint: In(a) = S"dx, please check the…

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Q: if x~unif. (0,2),find the U.b. of p(lx - mul>= 2)

A: Given X~Unif(0,2) Result: if X~unif(a,b) then  Mean=(a+b)/2 Variance=(b-a)2/12

Q: 12 3. If u lx + my, v = mx-ly, show that ди 4 + m

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Q: 5. prove thatS" erf (x) dx =o CA

A: ddx(erf(x)) =2 e-x2π   I =∫-aa erf(x) dxI= erf(x) x -∫ ddxerf(x) x dxI = x erf(x) -2π∫ x e-x2 dx-x2…

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

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Q: A large but sparsely populated county has two small hospitals, one at the south end of the county…

A: "Since you have posted a question with multiple subparts, we will solve first 3 sub-parts for you.…

Q: y If z = show that = 0. x + y?*

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Q: Show that For all z E C, with z # 0, then: |z/²z=1 = z

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Q: .Prove that e- -dt < (1+ t²)* 1/4 6.

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Q: alue of the pur C: |z| = (:-1)(z-

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Q: 6. Find a²z дгу of z = In(xy)

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Q: 2. If z = x° In 2.xy, find at 1,-

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Q: Find the fowien senies fon | X), -T<X <A

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Q: Suppose that x̅ = 12, Sx = 2, and z is 1. what is X?

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Q: S Prove that L(cos 3t) : %3D (s2+9)

A: We know that Lf(t) = ∫0∞e-stf(t) dt Then Lcos 3t = ∫0∞e-st cos 3t dt                             (1)…

Q: 2. Suppose the PDF of X is given by fx(x) = 5e-5x(o)(x). Find its CDF.

A: Solution

Q: 6. Use a substitution to find S (x* + 1)" z³dx.

A: Here we use substitution we find∫x4+111x3dxHere we substitute x4+1=du…

Q: X1,, Xn ~ N(-3,2) (random variables) ... Z, = 1(X,X2 + X1X3 + X1X4 + •..+ X1X„), n > 3 %3D (a) Find…

A: From the given information, X1, .....,Xn~N-3,2 And Zn=1nX1X2+X1X3+...+X1Xn, n≥3 Consider,…

Q: If Σ α An diverges then E ca can diverges only if c > 0 only if c < 0 only if c+ 0 for any real…

A: Given, ∑an diverges then find for which values of c, ∑can diverges.

Q: If Σ An diverges then > can diverges O only if c>0 O only if c< 0 O only if c+0 O for any real value…

A: Given, the series ∑an diverges, then we have to find the values of c for which the series ∑can…

Q: Prove that Z(Sn) = {e} for n ≥ 3.

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Q: Let Sn=Vn. Find an and show that an O but that Sn=ER=1@k diverges. ->

A: Solution:-

Q: 2. Solve the given SLE using LU-Factorization X, + x, +xz = 3 x, - x, +4x; = 4 2х, + 3x, — 5х, %3D0…

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Q: ronskian of the DE : xp(x)y' + 3y = 0 = er² et? et?

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Q: SSR is defined by SSR = E9 - 9). Prove %3D (1) SSR = 3} S.z- (2) E(SSR) = o + Bị Sr. (3) Show MSR ~…

A: Given, SSR=∑i=1ny^i-y¯2. The fitted equation of yi=β0+β1xi+εi is  y^i=β^0+β^1xi, for i=1,2,...,n. yi…

Q: b) = f ) then prove that x ux + y uy + z uz = 0 u

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Q: (2) If U = (1 – 2xy + y²)-³ prove that, x- = y?U® %3D

A: Since you have asked for a multiple question, we will solve the first question for you. If you want…

Q: 11. Prove the absorption law x +xy=x using the other laws in Table 5.

A: To prove: x+xy=x We use the following identities (1) x=x·1 (2) 1+x=1 (3) xy+z=xy+xz    ( This is…

Q: An M.V.U. is unique in the sense that if T, 1 M.V.U. estimators for y (0), then T, = T, almost

A: A sample statistics t is said to be unbiased estimator of population parameterγ(θ) if expected value…

Q: If (x* – y*,x – z°) = 0,z = z(x, y), prove: az y3 az +x³z4 ax - %3D ду 3

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Q: In order to obtain a good linear approximation for In (7.4) what would a particularly useful nearby…

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

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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: 8. Theorem h° = %3D

A: The figure given below represents the circle. From the angle subtended by the same arc theorem, The…

Q: If t,= 4n+1, find t,, t,, t3, and t Express t n+1, in its simplest form. n+1 In-1

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Q: Let Xi- N co,1) and n=2 auel T= Xi+ Xz Show that Tis a %3D Sufficiency Stabisuic Statistic.

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Q: In GL(2, Z,,). det

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Q: An M.V.U. is unique in the sense that if T, and T, are M.V.U. estimators for y (0), then T, = T,…

A: Answer: For the given data,

Q: 3. If f(x,y,z)= In 4 xyz Ite t3y:²s²) , find f,,f, and f..

A: Given that: f(x,y,z)=lnxyz4+e-(3xy2z2)

Q: Show that for all x, y E F, there exists e with e < Emachine such that fl(x * y) = (x * y)(1+ €) is…

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Q: IF Y = u²-1 u²+1 Find dy np

A: Given,  y=u2-1u2+1

Q: 13. If w(x,y) = In(1+xy²), show that 2wxx + y³Wyx = 0. %3|

A: Given: w(x, y)=ln(1+xy2)  To show: 2wxx+y3wyx=0 Chain rule : ddx(f(g(x))=f'(g(x))g'(x) Quotient…

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

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Q: If θ̂ is a point estimator of θ, prove that MSE(θ̂) = V(θ̂) + (B(θ̂))2

A: We have given that θ̂ is a point estimator of θ. We know that  MSE(θ̂) = E((θ̂ - θ)2 )