2. You are given the following multivariate PDF (x, y, z) E S X,Y,2(x, y,z)=)4m 0...
You are given the following multivariate PDF (x, y, z) ES fxx.2(x, y, z) =- 0 else where S-((z, y, z) 1x2 + уг + z2 < 1} (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inscribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S....
You are given the following multivariate PDF (z, y, z) ES else fxx,z(x, y, z) = ) 0 where S-((x, y, z) | x2 + y2 + z2-1) (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inscribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of s....
2. You are given the following multivariate PDF 3 (x, y, z) else s fxx.2(z, y, z)- I, 0 where S-((z, y,2)lr'ザ+8-1) (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inscribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S. What would the probabilities P(X,Y, Z)...
2. You are given the following multivariate PDF 0 else where S = {(z, y, z) |エ2 + y2 + 22-1). (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inseribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S. What would the probabilities P[(X, Y, Z)...
Problem 2 - Three Continuous Random Variables Suppose X,Y,Z have joint pdf given by fx,YZ(xgz) = k xyz if 0 S$ 1,0 rS 1,0 25 1 ) and fxyZ(x,y,z) = 0, otherwise. (a) Find k so that fxyz(x.yz) is a genuine probability density function. (b) Are X,Y,Z independent? (c) Find PXs 1/2, Y s 1/3, Z s1/4). (d) Find the marginal pdf fxy(x.y). (e) Find the marginal pdf fx(x).
Problem 2 - Three Continuous Random Variables Suppose X,Y,Z have joint...
Let S = {(x, y, z) e R: sin(fxyz) + €2-* = 1}. Show that there is a positive real number r such that B(p;r) nS is the graph of a Cl function, where p = (1,1,1).
1. Suppose that E(X) E(Y) E(Z) 2 Y and Z are independent, Cov(X, Y) V(X) V(Z) 4, V(Y) = 3 Let U X 3Y +Z and W = 2X + Y + Z 1, and Cov(X, Z) = -1 Compute E(U) and V (U) b. Compute Cov(U, W). а.
(1 point) Use spherical coordinates to evaluate the triple integral dV, e-(x+y+z) E Vx2 + y2 + z2 where E is the region bounded by the spheres x² + y2 + z2 = 4 and x² + y2 + z2 16. Answer =
(2) Let f(z, y)-xy +x-y be defined on the closed disk {(z, y) E R2 : z? + y2 < 4} of radius 2. (a) Find the maximu and minimu of Duf at (0,0) over all unit vectors u. (b) Find the maximum and minimum of Duf over all points in the disk(,y) E R2 r2 + y2 < 4} and all unit vectors u. (llint. Think of IvJF as a function ofェand y in the disk.)
2. Let X and Y be jointly Gaussian random variables. Let ElX] = 0, E[Y] = 0, ElX2-4. Ey2- 4, and PXY = [5] (a) Define W2x +3. Find the probability density function fw ( of W. [101 (b) Define Z 2X - 3Y. Find P(Z > 3) 5] (c) Find E[WZ], where W and Z are defined in parts (a) and (b), respectively.