Question

PROB5

Let U and V be independent r.v's such that the p.d.f of U is fu(u) = { 2 OSU< 27, otherwise.

Answer 1

Vzo Let x = vav cos U and Y = 12v sinu. Show that X and Y are independent Standard normal variables N101) Given that fulv) = 3v. 05 U£ 271 lo Otherwise fucw = feu. Uzo Juw l o otherwise fuv (u, v) = at euOzusät x = 1QV COSU, Y = Vav sin u X' = 2 v cos?u, Y = 2V sinau x2 + y2 = 9V -> V= x2 + y2 - Cot u => tan U= ) Then , 0405271, 04100 -15 COSU ST, 0% 12V Loo => X,Ye R Clearly the transformation (uv) >(x,y) is one-to-one from X-{luv): 044 £ 271, V20? 2

0 to DR Jacobicin iş J: Talu locksyj - 1-4/22 a I seciu / Ver sec²u I [: a = X and tan u = { 1 = sec u 24 12 7 1 x Seciu al y ] en seco Seciu secu ( +.)- - Secu (-1) =-{1tl4+) = -1 + U seczu 22* sec²u 22* Hence. fxy (x1y) = be 179) Het heef(xy) ER - fx (X). fx 10) # (x,y) ER? Hence, X and Y we independently distributed as W (0,0).