Let X and Y have joint density fx,y(x, y) = (6 - x - y) I(0,2)(x) I(2,4) (y). a. Find Var[Y| X = x]. b. Show that E[Y] = E{E[Y|X]}.

Q: Which of the following functions are continuous? 0. .(x, y) =(0,0) f(x,y) ={ 2x(x² – y²) I) (x, y) =…

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Q: The joint pdf of X and Y is given by fx,y(x,y) = {4x2   0<y<x<1}…

A: ANSWER: To determine the basis of a random variable U, we need to find the range of values ​​that U…

Q: Q.Let g: [1,4] → R be continuous function. Define G(x) = g(t)dt. Then dx dG a. G(x) is…

A: Here given g:1, 4→ℝ be a continuous function. Also, define Gx=∫ax2g(t)dt

Q: e /// 2 Evaluate 2xz dV where E {(x, y, z) |1 ≤ x ≤ 4, r ≤ y ≤ 2r, 0 <z<x+3y}

A:

Q: The joint pdf of X and Y is given by fx,y(x,y) = {4x2   0<y<x<1}…

A: Sol:- To find the support of U, we first need to find the range of values that U can take

Q: Let f(x,y) = and let g(x, y) 4xy¹ 4x4 + 4y4 0 if(x, y) = (0,0) 0 if (x, y) = (0,0) 5xy 10 5x² + 5y20…

A: f(x,y)=4xy44x4+4y4if (x,y)≠(0,0)0if (x,y)=(0,0)g(x,y)=5xy105x2+5y20if (x,y)≠(0,0)0if (x,y)=(0,0)

Q: Suppose f(x, y, z)=x+y+z, h(x,y) = (x + y, x³, y) and g(x) = (x,x²). Calculate d =z=of oho g(x). dx

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Q: // | *dædydzB = {(x, y, 2) =€ R® 0 <« < 1,0 < y < 1,0 < z < x} {(x, y, 2) =€ R° 0<æ < 1,0 < y < 1,0…

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Q: Let a 0 for all x = [a, b]. Let A > 0, B > 0 be constants. S[y] = is given by y(x) [* dx ƒ(x) √ 1 +…

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Q: Let f(x,y) = cx2y, x2<=y<=1, -1<=x<=1 be the pdf of X and Y. Find the value of c that makes this a…

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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: Suppose that f(x, y) = 7, and D = {(x, y) | x² + y² ≤ 1}. S f(x,y) dx dy =

A: Tou have to know the following result to solve this problem

Q: Given the continuous function: (a) f(x, y) = (b) fy(x, y) = (c) fax(x, y) = d) fxy(x, y) = f(x, y) =…

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Q: Evaluate 3xz dV where E {(x, y, z) | 2 < x < 5, x < y < 2æ, 0 < z < x + 2y}

A: Given,

Q: Suppose f is a differentiable function of x and y, and g(u, v) = f(e" + sin(v), eu + cos(v)). Use…

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Q: If f(x,y,z) = 2x² − yz + xz², where x = 2 sint, y = t² − t + 1, and z = e−³t, find f't)

A: from chain rule of derivaties…

Q: 6. Let p(t) = f (g (t) , h (t)), where ƒ is differentiable, g(2) = 1, g' (2) = –3, h (2) = 7, h' (2)…

A: p(t) = f ( g (t), h (t) g (2) =1 , g' (2) =-3h (2) =7 h' (2) =8 Fx (1,7) Fy (1,7) =8to find p'…

Q: /// 2xz dV where = {(x, y, z) | 1 < x < 4, x < y < 2æ, 0 < z < x + 3y} Evaluate E E %3D

A: We are asked to evaluate the integral.

Q: 5. The pdf of Y is f(y) = c/y³, y > 1. a) Calculate the value c so that f(y) is a valid pdf. b) Find…

A: Given probability density functionf(y)=c/y3,y&gt;1

Q: Let h(x, y) = x5 + y sin²x x² + y² 0, { 25 if (x, y) = (0,0) if (x, y) = (0,0) (a) Show that h is…

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Q: ey f(x, y, z) = 2x / -42³e" + In (x³-2). Let f (x Find fx, fy and fz.

A: Step 1: Find fx (Partial derivative with respect to x)To find fx​, we differentiate f with respect…

Q: Let f(x, y) = 6e sin(7y) af af dy

A:

Q: Determine k, so that ƒx,y(x, y) (x,y) =. a) Determine fylx=.25(0.5) b) Determine E[X] c) Determine…

A: Given: The joint pdf is given fx,yx,y=kx0&lt;x&lt;y&lt;10otherwise

Q: If f(x, y, z) = c²y² + ln(y+z), find ƒ₂(3,0,2), fy(1, 0, 2), and f(0,0,1)

A: We have to apply the partial differentiation to find solution of the given problem.

Q: Let f(x, y) = e-o-Y 0 < x < ∞, 0 < y < ∞ be the joint pdf of X and Y. Find the pdf and cdf of X and…

A: The joint probability density function is given as below: f(x,y)=e-x-y ; 0&lt;x&lt;∞, 0&lt;y&lt;∞

Q: If f(x) is continuous on a, b] thenF(x) = So f(t)dt is continuous. OTrue False

A: We need to check whether the given statement is true or false

Q: Let X and Y have the joint pdf f(x, y) = (1 – xy), 0 < x < 1, 0 < y < 1. Find E(X).

A: The given joint pdf is fx,y=431-xy,0&lt;x&lt;1,0&lt;y&lt;1.