a) Give a condition on two variables X and Y such that E[XY] = E[X]E[Y]. %3D Suppose that X ~U({-1,0,1}) and U ~U({-1,1}) are independent. Let Y XU. b) Show that X and Y are uncorrelated. c) Show that X and Y are not independent.

Q: Let fxy (x, y) = 6 (x – y) , 0 < y < x < 1. If E (X) = , E (Y) = Cov (X, Y) = then %3D

A: Given,fXY(x,y)=6(x-y) ; 0&lt;y&lt;x&lt;1E(X)=34E(Y)=14

Q: Consider the functions z= -6 ex In y, x= In (u cos v), and y = u sin v. dz dz (a) Express and as…

A: The given functions are z = -6ex lny,  x = lnu cosv, and y = u sinv. (a) The objective is to express…

Q: find c, and c2 so that y(x) = c1 sin x+ c2 cos x will satisfy the given conditions. Determine…

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Q: Consider the functions z =2e* In y, x In (u cos v), and y= u sin v. dz (a) Express and du dz as…

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

Q: Let x = s e and y =t es. Compute the Jacobian

A: Given: x=set and y=tes.

Q: D¹(²)y(x) + Dª(²)y(x − 0.01) + y¨ (x) (39 − y(x) = g(x), x = [0,h], solv using the subjected…

A: In natural philosophy and arithmetic, Legendre polynomials (named after Adrien-Marie Legendre, who…

Q: Let g(r, y) = ye-, r(t) = In(t² + 1), and y(t) = vt. Find g(r(t), y(t)).

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

Q: Consider the functions z = 4 e* In y, x= In (u cos v), and y =u sin v. dz and du dz (a) Express as…

A: Given, z=4exlny,  x=lnucosv,  y=usinv Part (a): z=4exlny,  x=lnucosv,…

Q: 6. Assume a joint pdf given as [λxy², 0≤x≤ y ≤1 fxx2,y)= (0, otherwise and let B = {Y≥ 2X and W=X+Y…

A: Hello! As you have posted more than 3 sub parts, we are answering the first 3 sub-parts.  In case…

Q: If z = f(u) + g(u), where u = - and v = x y. Prove that y %3D x: Zx - y· Zy = y2 · Zyy – x2 . Zxx…

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

Q: Show that: (a) Cov(X,Y) = E[XY] – E[X]E[Y]. (b) If X and Y are independent, then fY|X(y|x) = fY(y)

A: Solution:a.Show that COV(X,Y)=E(XY)–E(X)E(Y)

Q: Given that vec F =x^ 2 y hat i -(z^ 3 -3x) hat j +4y^ 2 hat k find the k- component of ( nabla* vec…

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Q: Prove that if a e R, then a 0 = 0 = 0 · a. Suppose x, y, z E R. Show that xz = yz if and only if…

A: We will prove the given statements.

Q: * Let A E R"Xn, ĩ E R" \ {0} and A = 0. Prove that if the origin is stable if and only if ī is…

A: Given A∈ℝn×n,x¯∈ℝn\0 and Ax¯=0 To prove that the origin is stable if and only if x¯ is stable.  If…

Q: If z = f (u) + g(v), where u = = and v = x ·y. Prove that Q1: x· Zx – y· Zy = y² · Zyy – x² · Zxx…

A: We have z=fu+gv, and u=xy  and v=x.y We need to use the chain rule and prove x.zx-y.zy=y2.zyy-x2.zxx…

Q: Let f(x, y, z) = ln(x² + z) + y²e®z – cos(2yz). Find fzyn. Zyx ·

A: Given function is  f(x, y, z)=ln(x2+z)+y2exz-cos(2yz)

Q: Looking Ahead Use the Wronskian to show that {e' , te' I is linearly independent on R.

A: Consider the equation

Q: Given that y, y, be a fundamental set of solution to the D. E. (ey)' + y + y = 0, x > 0 If the…

A: The solution is given as

Q: Let u = cos(r + y), x = 2st +t and y = t – 2s. Use the chain rule theorem to compute du for du and…

A:

Q: Find and ди ди using the chain rule. w(x, y) = 14x²y², x = e" cos v, (u, υ) ди (u, υ)

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Q: Let z = = g(x, y) = e* cos y +y In(x – 1).

A: Let we havez=g(x,y)=excosy+yln(x-1)If x=s-t and y=s2-t2

Q: Show that the functions e*, cos x, and sin x are linearly independent using the Wronskian.

A:

Q: Let f(x, y) xy, x = 1- Vt, and y = 1+ Vt. Find df using the chain rule. dt

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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: Determine if the functions Y1=e^{4x}senx and Y2=e^{4x}cosx are linearly independent or linearly…

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Q: A function f(x,y) is homogeneous of degree 7 in x and y if and only if, __________.

A: Given problem is solved below.

Q: ; Let x, y ER" be given. (a) lf ||x|| = ||y|| = 1 and x - y = 1, show that x = y. (b) If ||x|| ≤ 1…

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Q: 7. Suppose that f (x, y) = e" cos TY and u? – v², y = u? + v?. x = Using the chain rule, compute the…

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Q: Suppose that w = f(x, y, z), x = h(s), y= g(s, t), z= k(s,t) and all the conditions required for the…

A: By using the chain rule for partial differentiation we will find the value of it.

Q: a When reduced to a linear DE of order one in z, the Bernoulli's equation dx becomes dz A + z( − 1)…

A:

Q: e) Suppose that W= √√x² + y² + z², x= cos 0, y = sin 0, z = tan 0. dw T Use the chain rule to find…

A: According to our company's guidelines I can only answer (e), since (e) and (f) are not inter…

Q: If z = f(u) + g), where u = = and v = x· y. Prove that y x: Zx - y Zy = y? Zyy – x2 - Zxx using the…

A: We have to prove this

Q: 2. Use the Chain Rule to find for z = e¤/y where x = st and y = s2 +t

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Q: Determine DFT X[ej"] of the signal (answer a or b) a) x[n]coswon. b) x(n) = a"u(-n- 1) %3D

A: As it's mentioned in the question to answer between a) or b), so showing steps for only one part. b)…

Q: Let z, w E C. Show that zw = 0 implies z = 0 or w = 0. %3D

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Q: SSS zdV Q3: Evaluate E + y² + z² = 1 and x² + y² + z² = 4 in the upper where E lies between the…

A: We have to evaluate the integral ∭EzdV, where E lies between the spheres x2+y2+z2=1 and x2+y2+z2=4…

Q: Express dw/ar and aw/as as functions of r and s using chain rule w = In(x² + y² + z²), x = re sin r,…

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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: Given that y1, Y2 be a fundamental set of solution to the D. E. (e*y')' + y' + y = 0, x > 0 If the…

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Q: Let {x; t = 1,2, ...} be a covariance stationary process and define y, = Cov(x, x+h) for h = 0.…

A: It is given that {xt: t = 1,2,…} is a covariate stationary process. It implies that, E(xt) and V(xt)…

Q: For x = 1, 2, 3 and y = 1, 2, let the joint pmf of X and Y be defined by fX,Y (x, y) = x + y/21…

A: Solution

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: Let w f(t,0) where t and 0 are functions ofr and s, %3D dw is equal to Or then according to the…

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Q: that X and Y have joint pdf f(x, y) ={x + 3x^2y if 0 <= x <= 1 and 0 <= y <= 1, 0 otherwise.   (c)…

A: From the given information, Consider, the Joint pdf of X and Y is as follows: fx,y={x+3x2y if…

Q: The joint PDF of X and Y is given by f (x, y) = x+ 0 0 and y E (0, 1), derive E[X | Y = y] in terms…

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Q: Let z = f(x, y) where g = e° cos t and y = e° sin t. Use the Chain Rule to show: z = e

A: Let z = f(x, y) where x = es cos t and y = es  sin t. We will be using the chain rule to show,…

Q: The values of m and k that makes the ODE (yx+ )d ym = x*(ny)-In(x*) dy %3D homogeneous are m=-3, k=4…

A: Given ODE is said to be Homogeneous if they can put in the form dydx=fyx . Since we have…

Q: (a) Let F z sin y, where r(s,t) = st and y(s, t) = st. Use the chain rule to find %3D %3! OF

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Q: 6. Let f(u, v) = u sin(v) – v cos(u) and u(r,y) = r² – ry² v(r, y) = y – r Find af af and both by…

A: We need to find the derivatives using direct substitution and by chain rule . Now , we have ,…