Question

9.155 Let the location of a particle moving in two dimensions be described by a Cartesian coordinate system whose origin is the location of the particle at time t = 0. A common model for this process is that the x and y coordinates of the location of the particle at time t > 0 are independent N(0,02t) random variables. Let R(t) and (t) denote the polar coordinates of the position of the particle at time t > 0. For fixed t > 0, a) find a joint PDF of R(t) and (t). b) obtain and identify marginal PDFs of R(t) and (t). c) decide whether R(t) and (t) are independent random variables.

Answer 1

& and y follows (0,0²%) independenty and to fixed I where - NexL R(E) and o(6) are the polar sayes I co-ordinate St is fixed asume R(4) = 010-0 :: x=rcoso y = rsino x + y² = 82 I Losorsino J: 0(x,y) 0(2.0) sino rcoso :: 1J1 -6 8o and now OLOCZT a) :66,0 (1.0). TREERS 5) f (P) S do

fo- * 5 L Ext { het nie of - 7 nou = 27 stor Lenjerie falo) re 660-, cines () clearly fr® (6,0)= FR (6) x 6 0 there they 227 are independent.