Question

A continuous probability density fanction is a non-negati ve continuous function f with integral over its entire domain D Rn equal to unity. The domain D may have any number n of dimensions. Thus . . .lofdェ1 . . . drn-1, The mean, also called expectation, of a function q is denoted by尋or E(q) and defined by 1··· DG-f) d工1-.. drn. The same function fmay also represent a density of matter or a density of electrical charges Definition 1 The Bivariate Cauchy Probability Density Function f is defined over the whole plane D R2 by Proble To verify that fis a probability density function, calculate (A similar integral appears as an exercise in the calculus text [ 1, Ch 14, § 14.3, Exercises 66, p 989). (1.2) To locate the mean (T, of f, calculate, or determine by mathematical considerations (1.3) To deter mine whether the variance exists, calculate d A T2 + The probability that (r, y) fall in a subset E of the plane is defined by Problem! 2 (2,1) Calculate P[D(0. R)] for E = D(0, R), which is the disc with radius R centered at the origin: D(0, R) = {(z, y) E R2 :ェ2 + y2 < R2} (2.2) Calculate PAS R)] for AS R) = {(z, y) E R2 : S2 <ェ2 + y2 < R*), which is called an annulus (2.3) Calculate P(H' ) for E-H+-{(z, y) E R2 : y > 0), which is called the upper half plane. (Think!) (2.4) Calculate P(Sam) for Sla,b] = [a,&J × R = {(z, y) E R2 : a *S b), which is a vertical strip. (25) Calculate P(R|:::|) for R1.1-[a,#x [c,4-((z, y) E R2 : (a *S b) AND (c < y d)) dA. Problem 3 Parametrize the unit sphere S2 with a stereographic projection: a stereographic map of a spherical Earth with the South Pôle in Antartica at the origin. If D R2 is mapped to.AG S2, then I A f dA-IO?

Answer 1

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