Suppose Y₁,..., Yn are i.i.d. random variables with Y; ~ N(u, o). Express the following vector in the form Y Y₁ - Y₂ Y2 - Y3 AY and find its mean and variance: Yn-1 - Yn
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- The distribution of a random vector is given by a function:f(x,y) = y/2 - x/2 ; for 0 < x < 1 and 2 < y < 3 = 0 ; otherwise Determine P( X > Y - 2), and sketch the graf.Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)Let continuous bivariate random vector (X, Y ) have a jointly uniform distribution on the unit square:
- (a) show that the set of vectors in Rn is orthogonal, and(b) normalize the set to produce an orthonormal set.{(√3, √3, √3), (−√2, 0, √2)}Let X and Y be two continuous random variables having joint pdffX,Y (x, y) = (1 + XY)/4, −1 ≤x ≤1, −1 ≤y ≤1.Show that X ^2 and Y ^2 are independent.What will happen if the Gram–Schmidt process is applied to a set of linearly dependent vectors {v1, v2, v3} such that v1 and v2 are linearly independent, but v3 ∈ Span(v1, v2)? Will the process fail? If so, how? Explain.
- the normal vector v1 should be (1,3,-1)Random vector X has PDF fX(x) = {ca'x when 0≤x≤1 and 0 otherwise} where a=[a1,...,an]' is a vector with each component ai > 0. What is c?Let random variables X and Y have the joint pdf fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1 0, otherwise Find the joint pdf of U = X^2 and V = XY.