Question

C Nievergelt New Homework on Cauchy 's 2D Density Function due Thurday 23 May 20191 2019 Y. A continuous probability density function is a non-negative continuous function fwith integral over its entire domain D S R equal to urity. The domain D may have any number n of dimensions. Thus .Jpfdri..d -L The mean, also called expectation, of a function q is denoted by q or E(q) and defined by q .. (q-fJdr. , .dh The same functionfmay 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 R by Problem 1 To verify that fis a probability density function, calculade (A similar integral appears as an exercise in the calculus text [l. Ch. I4, § 14.3, Exercises 66, p. 989D. (1.2) To loc ale the mean ,of f. calculate, or determine by mathematical considerations +r+y (1.3) Todetermine whether the variance exists, calculate 司":=氵.lla.(z .. Tァー·(1+ r +7) dA The probability that (a,y) all in a subset E of the plane is defined by Problem 22.1) Calculate PDIO,R)for DOR), which is the dise with radius R centered at the origin: (2.2) Calculate Plas. R〗 for A(SR)-{(z.v) E R2 . S2くみ/c R, which is called an angulus. (2.3) Calculate P(H+ ) for8. + s((z. y) E R2 : > 0), which is called the upper half plane mink!) (2.4) Calculate P(Sa,bi) fors,测-la.q x R-((z,y) € R2 : a b). which is a vertical strip. エ Problem 3 Parametrize the unit sphere S with a stereographic projection: a stereographic map of a spherical Earth with the South Pôle in Antartica at the origin. I DC R2 is mapped t AS then A dA Ip? References Willam Briggs, Lyke Cochsan.and Bemard Gilks. Calcfas. Pearson Baste.MA secoad edton, 2016. With e assistance f Enic Scul ISBN 978-0-321-95435-0 Q4303 21 749 2015 S15-d23 LOCC No 2003019980

Answer 1

iven that anti nous Prohability density function non ← negative Continous fundion f wHt Mtegradou change to pola Co-oRdivatex Hence we ge, 2. Integ ration oueR &

dt A び Wantshes 2 Sino ido

To detemine whethes the vaviance Smiste 2. Oro 1-tu blows up because first fein

blaus up beause 익 fint "team ence the

As per HomeworkLib policy we need to solve one question per post. Please post the remaining questions in another post.