TS (Y V is a two-dimensional RV uniformly distributed over the triangular region R bounded by y = 0, x = 3 and y = 4/3 x. Find f,(x), f,(y), E(X), Var(X), %3D E(Y), Var(Y) and Pry
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- Suppose that (X,Y) is uniformly distributed over the region {(x,y): 0 < |y| < x < 1}. Find E(X) and E(Y).integrate ƒ over the given region. ƒ(x, y) = x2 + y2 over the triangular region with vertices (0, 0), (1, 0), and (0, 1)Integrate f(x, y, z) = x over the region in the first octant bounded above by z = 8 - 2x 2 - y2 and below by z = y2 .
- Integrate f (x, y) = x over the region bounded by y = x, y = 4x − x^2, and y = 0 in two ways: as a vertically simple region and as a horizontally simple region.Integrate f(x, y) = x over the region bounded by y = x 2 and y = X + 2.integrate ƒ over the given region. ƒ(x, y) = 1/(xy) over the square 1<=x<= 2, 1 <=y<=2.
- Suppose that (X,Y) is uniformly distributed over the region {(x,y): 0 < |y| < x < 1}. Find: a) the joint density of (X, Y) b) the marginal densities fx(x) and fy(y) c) Are X and Y independent? d) Find E(X) and E(Y).A health-food store stocks two different brands of a certain type of grain. Let X = theamount (lb) of brand A on hand and Y = the amount of brand B on hand. Suppose thejoint pdf of X and Y isf(x, y) =kxy x ≥ 0, y ≥ 0, 20 ≤ x + y ≤ 300 otherwisea. Draw the region of positive density and determine the value of k.b. Are X and Y independent? Answer by first deriving the marginal pdf of eachvariable.c. Compute P(X + Y ≤ 25).d. What is the expected total amount of this grain on hand?e. Compute Cov(X, Y) and Corr(X, Y).f. What is the variance of the total amount of grain on hand?Find the maximum and minimum of w(x, z) = 2xz + z^2 +1 in a region R bounded by x = 0, z = 0, and x + z = 1(in the first quadrant).
- find the average value of F(x, y, z) over the givenregion. F(x, y, z) = x + y - z over the rectangular box in the first octant bounded by the coordinate planes and the planes x = 1, y = 1, and z = 2The graphs of f (x) = x2 and g (x) = cx3 , being c> 0, intersect at the points (0,0) and at (1 / c, 1 / c2 ). Determine c such that the bounded region between these graphs and on the interval (0, 1 / c) has area 2/3