4) Let xn +1 =- + rzn for r > 0. (A) Find the fixed points (in terrns of r) and use the derivativ...
7. Consider a family of maps f :R-R, where f(r)= 2+c, cE R. a) Let c 0. Find all the fixed points of f and analyze the map by drawing a cobweb. Check stability of the fixed points b) Find and classify all the fixed points of f as a function of c. c) Find the values of c at which the fixed points bifurcate, and classify those bifurcations. d) For which values of c is there an attracting cycle...
4. (20 pts) Let {xn} be a Cauchy sequence. Show that a) (5 pts) {xn} is bounded. Hint: See Lecture 4 notes b) (5 pts) {Jxn} is a Cauchy sequence. Hint: Use the following inequality ||x| - |y|| < |x - y|, for all x, y E R. _ subsequence of {xn} and xn c) (5 pts) If {xnk} is a See Lecture 4 notes. as k - oo, then xn OO as n»oo. Hint: > d) (5 pts) If...
2-3. Let ?>0 and ?? R. Let X1,X2, distribution with probability density function , Xn be a random sample from the zero otherwise suppose ? is known. ( Homework #8 ): W-X-5 has an Exponential ( 2. Recall --)-Gamma ( -1,0--) distribution. a) Find a sufficient statistic Y-u(X1, X2, , Xn) for ? b) Suggest a confidence interval for ? with (1-?) 100% confidence level. "Flint": Use ?(X,-8) ? w, c) Suppose n-4, ?-2, and X1-215, X2-2.55, X3-210, X4-2.20. i-1...
2) Difference equations of the form xn+1=ƒ(xn) are a somewhat old-fashioned approach. A more modern approach is to use difference equations of the form Δx=ƒ(xn) where Δx = xn+1- xn. Explore the difference equation Δx=ƒ=r xn (1-xn). a) Algebraically determine the equilibrium and its stability as functions of r. b) Construct a simple bifurcation by hand by graphing the equilibria as a function of r. Represent a stable equilibrium with a solid graph and an unstable equilibrium with a dashed or...
8(100) Let X1,,Xn be iid from r(a, 6). (1)(50) Find the limiting distribution of the MLE of B. (2)(30) Find the limiting distribution of the MLE of B when a is known. (3)(20) Compare two asymptotic variances in (1) and (2), and make comment on it. 1ラ 8(100) Let X1,,Xn be iid from r(a, 6). (1)(50) Find the limiting distribution of the MLE of B. (2)(30) Find the limiting distribution of the MLE of B when a is known. (3)(20)...
4.(120) Let X1,,,Xn be iid r(, 1) and g(u) given. Let 6n be the MLE of g(4) (1)(60) Find the asymptotic distribution of 6, (2)(60) Find the ARE of T Icc(X) w.r.t. on P(X1> c), c > 0 is i n i1 5.(80) Let X1, ,,Xn be iid with E(X1) = u and Var(X1) limiting distribution of nlog (1 +). o2. Find the where T n(X - 4)/s. - 1 - 4.(120) Let X1,,,Xn be iid r(, 1) and g(u)...
I. Let 2r2y"-ry' + (r' + 1) y = 0. ) Verify that a 0 is a regular singular point. (2 points) ) Find two linearly independent power series solutions about a0. Provide at least3 aon-zero terms of each solution. (8 points) 0 0 is reowlar 터 -2 kti I'm Stuck-HEKG-HER- ME FINISH I. Let 2r2y"-ry' + (r' + 1) y = 0. ) Verify that a 0 is a regular singular point. (2 points) ) Find two linearly independent...
i d 9. Let Xi . . . , xn Uniform(9.0+1), θ R. Show that the minimal sufficient statistic T (X(1), X()) is not complete. Hint: use the results in Example 6.2.17 of Casella of Berger (2002) i d 9. Let Xi . . . , xn Uniform(9.0+1), θ R. Show that the minimal sufficient statistic T (X(1), X()) is not complete. Hint: use the results in Example 6.2.17 of Casella of Berger (2002)
a) Interaction between two interaction between the species is described by the following system. (A) fixed points. Assume 0 and y 2 0. (only 1st qundrant) species. Let r and y represent the populations of two species. The assify the Find and el (B) Sketch the phase portrait. Show only the 1st quadrant. Include nullelines. (C) Interpret your results in terms of the two species. a) Interaction between two interaction between the species is described by the following system. (A)...
Likelihood Ratio Tests - I only require (a) and (b) here. I'll post (c) and (d) for another question Let X1,..., Xn be a random sample from the distribution with pdf { 0-1e--)e f(r μ, θ ) - 0. where E Rand 0 > 0 (a) If 0 is known but a is unknown, find a likelihood ratio test (LRT) of size a for testing Η : μ> Ho Ho Ho versus where oi a known constant (b) If 0...