Question

A continuous probability density function is a non-negative continuous function f with integral over its entire domain D R" equal to unity. The domain D may have any number n of dimensions. Thus Jpfdzi..d 1. The mean, also called expectation, of a function q is denoted by or E(a) and defined by J.pla f)dy...dr The same function f may 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 f(r, y) Problem(1) To verify that fis a probability density function, calculate (1.2) To locate the mean (T, of f,calculate, or determine by mathematical considerations dA (1.3) To determine whether the variance exists, calculate 1+2 +y R2 dA Definition 2 The probability P() that (r,y) fall in a subset & of the plane is defined by (just a name for) the integral Problem 2 2.) Calculate P[D(O, R) for &-D(O, R), which is the disc with radius R centered at the origin (2.2) Calculate PlAS R) for A( S, R) = { (z.ye R2 . S2 < + rz < R2}, 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(Sla阎) for S[ -|a,&J × R = {(zw) € R2 : a Sr b). which is a vertical strip (2.5) Calculate P(RE:I) for R::: [a,#x le, d-{ (z, y) E R2 : (aS *S b) AND (c y d)). (Hard.)

Answer 1

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