Question

1. If U₁, U₂, U₃ are i.i.d. Unif(0,1), what’s the distribution of ?

2. If U and V are i.i.d. Unif(0,1), what’s the distribution of + ?

-3ln(U1(1- U2)(1 - U3)) -2 cos(2TV)-In(U)) n(U) sin(2T V)

Answer 1

U,, U, U ae fid OntP Co, Ci-U,) N Unif Co,) CrU unif Co,D >idCcNCH, ave . Let y,-30 0 nu, 3 es U dy 2-Rin 20 0-3n U,R dy Lo N Exponertial cth mean 3. Surilasy n-)Expmean -3) An C-,)~ ExP(mean 3) ase iid .Y,, Y2, Y ade also iid S;nce u,, 1-Up, 1-U2 fr,C3,)- ey, coc Gamma( 1) y, / Indegendkntly Gamma Cs, Y2 N Gamma by acditfve poogesty of Gamma distibution C 1tlt13) So ZEY,t Y2+ Y2 Gamma me CS Scanned with CamScanner5y 2 Tzro

U,v ase id aith unif C0,) Let, x-2 Inu sin env 0U,V ya-eAnu CoS 21 V fuvcuv I eu,ve) Oc-21nue -) Sin 21V 2 ) Anuo 1 e Cos 217 V - ny in eT V CoS 2nv xty21n u X Ftan env →V=レtanフx Xy2) X -e Y -I.X y2 2TT Y 1t2 2TT メ+ y2 20 e 2 -xX, y e N CO, 211 Similaly, ty a) N CO, リ XY NCO,り、 COS 2TT V -Any sin2TTV 24

NP-Anu sin 2TV + --2+ N2inu CoS 271V elnu sin eTV 2 +-2Anu cos 2TV No w XN NCC,D Y NNCO, 1id X+Y N Co, 1+D ) K+y N Co, Lt) N Co, ) N 7 NC-2, D -2+in Cos eT V +-Anu Sn enTV NNC2,