Consider the right triangle shown, where , y, and L are lengths. Starting each part of...
Consider the right triangle shown below that has side lengths of z, y and r units. 2 For each of the following questions, write an expression in terms of that a, y is how many times as large as r? wers the question. Enter "theta" for . times as large Preview b. r is how many times as large as r? times as large Preview c. y is how many times as large as z? times as large Preview Submit
Consider X1,X2, , Xn be an iid random sample fron Unif(0.0). Let θ = (끄+1) Y where Y = max(X1, x. . . . , X.). It can be easily shown that the cdf of Y is h(y) = Prp.SH-()" 1. Prove that Y is a biased estimator of θ and write down the expression of the bias 2. Prove that θ is an unbiased estimator of θ. 3. Determine and write down the cdf of 0 4. Discuss why...
(3) For the closed-loop system shown in the figure below, K varies from 0 to to. L(s) has 4 poles and no zero, and all the four poles are on the left half plane. Consider the root loci of L(s) for K approaching to too: how many branches of the root loci will be in the right half plane? (2 point) And Why? (2 points) L(s) Y(s) R(s) +
(3) For the closed-loop system shown in the figure below, K...
L(a) A long (L>> a) cylinder in vacuum has a line charge density p is shown below, (). State the Gauss's law for electric field in words. [1) (i). In order to calculate the electric field inside the cylinder using Gauss's Law, draw an appropriate Gaussian surface in the cylinder. [1] (i). Use the above information or otherwise, show that the electric field in the radial direction Pt inside the cylinder is ,2a (assume that the charge is evenly distributed...
Consider a cylinder of mass M, radius R and length L. (a) Calculate the inertia tensor for rotations about the center of mass in the frame where the z axis is along the axis of the cylinder. Use cylindrical coordinates, where x = r cos θ and y = r sin θ. (b) Find the inertia tensor in the frame where the center of the “bottom side” is at the origin with the z axis along the axis of the...
4. Suppose that Z є R2p has a MVN distribution Nop( 12. E.). Partition Z as Z where X є Rp and Y є Rp. Denote the means of X and Y as μι and μy, respectively. Let μΔ-Ha-ty. Suppose that we obtain IID data Z1, ,Zn from the underlying distribution of Z. Let a (0,1) be a constant (a) Describe how to construct a (1-a)-level convex confidence region (CR) for μΔ when y, is known. Explain. (b) Describe how...
5. Consider a long rectangular "gutter" of length a in the x direction and infi- nite height in the y direction. The gutter is infinitely long in the z direction, so the potential V inside the gutter only depends on x andy. The left (x-0,y), and right (r- a,y) sides of the gutter are grounded so that the potential V(x,y) is zero on those surfaces. The bottom surface of the gutter is kept fixed at a potential given by V(r,y-0)-...
1. This quasi-"walkthrough" problem is great practice in cross-products, vectors, and integration. Consider the current loop shown in Figure 2, with B _Bx, and the loop lying in the x - z plane of the page (y points into the page). We wish to find the net torque on this current loop. we'll do this by integrating in θ, the angle shown (a) We'll start with a little segment dl at point p as shown in the figure. What is...
2. RFID Tag Magnetic field: Consider a square loop of wire that lies in the x-y plane and carries an electric current lo. The center of the loop is located at the origin and each side has length a. The current flows in a counter-clockwise direction as shown in the figure below Note*: This is a common design for an RFID tag's antenna, we will analyze RFID tag detection at a later time. a) Using Biot-Savart's law, find an expression...
4. Suppose that Z є R2p has a MVN distribution M2p(Hs, z). Partition Z as Z- where X є Rp and Y є Rp. Denote the means of X and Y as μζ and Ily, respectively. Let μΔ As-ty. Suppose that we obtain IID data Z1, ,Zn from the underlying distribution of Z. Let a E (0,1) be a constant. (c) Describe how to construct a simultaneous confidence interval for νΤμΔ for all non- zero v є Rp when 31,...