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- 5.) Suppose the joint density for X and Y is given by f(x, y) = 1/xfor 0 < y < x < 1. Find ρX,YLet x denote the amount of time for which a book on 2-hour reserve at a college library is checked out by a student, and suppose that x has density func- tion f(x)5.5x for 0,x,2.Show that if α > 1 and β > 1, the beta density has arelative maximum atx = α − 1α + β − 2.
- Let X and Y be independent and exponential with parameter lambda. Show that the density of X conditional on X+Y=v is uniform on [0,v]8. Show that the gamma density function f(x, λ, r) integrates to 11.Suppose that f : [0, 1] −→ [0, 1] is a continuous function. Prove that f has afixed point in [0, 1], i.e., there is at least one real number x ∈ [0, 1] such thatf(x) = x. 2.The axes of two right circular cylinders of radius a intersect at a right angle.Find the volume of the solid of intersection of the cylinders.
- what local linearizatoin of the function f(x,y) sin(xy) + cos(x/y) at (pie/4, 1) (show all steps please)Verify divergence theorem for f=x2i+zi+yzk taken over the cube bounded by x=0,x=1,y=0,y=1,z=0 and z=13 Show that the square integrable function f(x) = sin( πk log x/ log 2 )for k ≥ 1 are orthogonal over the interval 1 ≤ x ≤ 2 with respect to the weight function r(x) = 1/ x . Obtain the norms of the functions and construct the othornormal set.
- Maximum divergence Within the cube {(x, y, z): | x | ≤ 1, | y | ≤ 1, | z | ≤ 1}, where does div F have the greatest magnitude when F = ⟨x2 - y2, xy2 z, 2xz⟩?Consider the unit cirlce in the plane; that is, the set of all points within a distanceof 1 of the origin: A= {(x, y) : x^2 + y^2 ≤ 1}.Suppose we choose a point (X, Y ) uniformly at random from S. What is the pdf for thedistance B = √X^2 + Y ^2 from the origin to this point? Hint given: What is the cdf of B?(The Second Derivative Test) Let f : [a, b] → R be differentiable on (a, b). Suppose c ∈ (a, b) is such that f '(c) = 0, and f ''(c) exists. (a) If f ''(c) > 0, prove that f has a local minimum at c. (b) If f ''(c) < 0, prove that f has a local maximum at c. (c) Show, using two specific examples, that no conclusion can be made if f ''(c) = 0.