6. Define a relation on the plane by setting (xo, yo) < (x1, yı) if either...
= Yo, Find the solution to the interpolation problem of finding a polynomial q(x) with deg(q) < 2 and such that q(xo) q(x1) = yi, and q'(x1) = yi with Xo < X1. Under what exact conditions is deg(q) = 2?
(a) Show that the points (x1, yı), (X2, y2), ..., (xn, yn) are collinear in R2 if and only if 1 X1 yi X2 Y3 rank < 2 1 Xn yn ] (b) What is the generalization of part (a) to points (x1, Yı, zı), (x2, y2, 22), ...,(Xn, Yn, zn) in R'. Explain.
3) Define the relation <on R via x < y if and only if xy < 10. Show that is symmetric. (20 points)
29. (a) Without solving, explain why the initial-value problem dy dx vy, y(xo) = yo has no solution for yo < 0. (b) Solve the initial-value problem in part (a) for yo > 0 and find the largest interval / on which the solution is defined
4. Let X1,..., X, be a random sample from a population with pdf 0 otherwise Let Xo) <...Xn)be the order statistics. Show that Xu/Xu) and X(n) are independent random variables
Find the solution to the interpolation problem of finding a polynomial q(a) with deg(q) < 2 and such that q(20) = yo, q(x1) = yi, and q' (x1) = y; with Xo < X1. Under what exact conditions is deg(q) = 2?
Let Xo, X1,... be a Markov chain with transition matrix 1(0 1 0 P 2 0 0 1 for 0< p< 1. Let g be a function defined by g(x) =亻1, if x = 1, if x = 2.3. , Let Yn = g(x,), for n 0. Show that Yo, Xi, is not a Markov chain.
Q6 (4+3+3+ 6=16 marks) Let Xo, X1, X2 be three distinct real numbers. For polynomials p(x) and q(x), define < p(x),q(x) >= p(xo)q(x0) + p(x1)q(x1) + p(x2)q(22). Let p(n) denote the vector space of all polynomials with degree more no than n. (i) Show that < .. > is an inner product in P(2). (ii) Is < ... > an inner product in P(3)? Explain why. (iii) Is <,:> an inner product in P(1)? Explain why. (iv) Consider Xo =...
2y (9 points) Given the initial value problem y' => y (xo) = yo. Use the existence and uniqueness theorem to show that a) a unique solution exists on any interval where xo + 0, b) no solution exists if y (0) = yo # 0, and c) an infinite number of solutions exist if y (0) = 0.
For i 1, let X G1/2 be distributed Geometrically with parameter 1/2 Define Xi -2 Approximate P (-1 < -2) with large enough n.