4. (3 pts) Two points are picked independently and uniformly from the region inside a unit circle...
Two points are picked independently at random from the region inside a circle. Let (R1) and (R2) be the distances of these points from the center of the circle.Find P((R2)(R1)/2).
Assume three points, one of which is red, are uniformly and independently distributed over the circum ference of a circular track, which has a total length of 1. Equivalently, you can interpret the circle as a unit interval with its two ends tied together, and all distances of points are measured in terms of the length on the track. The Figure below gives a simple illustration. 1. Let X be the distance of the red point to its neighbor in...
3. In a Monte Carlo method to estimate T, we draw n points uniformly on the unit square [0, 1]2 and count how many points X fall inside the unit circle. We then multiply this number by 4 and divide by n to find an estimator of T (a) What is the probability distribution of X? b) What is the approximate distribution of 4X/n for large n? (c) For n- 1000, suppose we observed 756 points inside the unit circle....
4. (15 points) Inside the region of R2 given by r2+y2 < 3 and x > 1, find the rectangle of maximal area whose sides are parallel to the x and y axes
From a given triangle of unit area, we choose two points independently with uniform distribution. The straight line connecting these points divides the triangle, with probability one, into a triangle and a quadrilateral. Calculate the expected values of the areas of these two regions.
4) A very LONG hollow cylindrical conducting shell (in electrostatic equilibrium) has an inner radius R1 and an outer radius R2 with a total charge -5Q distributed uniformly on its surfaces. Asume the length of the hollow conducting cylinder is "L" and L>R1 and L>> R2 The inside of the hollow cylindrical conducting shell (r < R1) is filled with nonconducting gel with a total charge QGEL distributed as ρ-Po*r' ( where po through out the N'L.Rİ volume a) Find...
3. [10 pts.] Evaluate the tripe integral //.Vz?ザ+zav where Eis the solid tripeintegr:=2.VETTFw where Eisthesolidball bounded by the sphere 2 + V2 + ? 4. [5 pts] Consider the region D, outside the circles Ca and Cs and inside the circle Ch in the Rt . d7-6芦. d7--2T, and Qz-P, = 2 on an open region containing D. Use figure below and a vector field F(, y)- (P(x, y), Q(x, y)). Assume we know that Cz dア Green's Theorem to...
3. [10 pts.] Evaluate the tripe integral //.Vz?ザ+zav where Eis the solid tripeintegr:=2.VETTFw where Eisthesolidball bounded by the sphere 2 + V2 + ? 4. [5 pts] Consider the region D, outside the circles Ca and Cs and inside the circle Ch in the Rt . d7-6芦. d7--2T, and Qz-P, = 2 on an open region containing D. Use figure below and a vector field F(, y)- (P(x, y), Q(x, y)). Assume we know that Cz dア Green's Theorem to...
7. (15 points) Let Xi and X2 be the position of two points drawn uniformly randomly and independently from the interval [0, 1]. Define Y = max(X,Xy) and Z-X1 + X2. (1) Calculate the joint PDF of Y and Z. (2) Derive the marginal PDF of both Y and Z. Are Y and Z independent? 7. (15 points) Let Xi and X2 be the position of two points drawn uniformly randomly and independently from the interval [0, 1]. Define Y...
4. (15 points) Suppose X1, X2, X are iid Bernouli (p) random variables, where pE (0,1) is unknown. Suppose we are testing by using the rejection region 81 C-(r1, r2, 3) :R 128 where R- and fo and fı are the joint probability functions under the null and alter- native hypotheses, respectively. a. Find the distribution of R under the null and alternative hypotheses. b. Find the expected value of R under the null and alternative hypotheses C. What is...
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