Question

Unif (0, 1) 5. Suppose U1 and U2 i= 1,2. Let X; = - log(1 - U;), i = 1,2. [0, 1], U are independent uniform random variables on (a) Show that X1 and X2 are independent exponential random variables with mean 1, X; ~ Еxp(1), і — 1,2. (b) Find the joint density function of Y1 = X1 + X2 and Y2 = X1/X2 and show that Y1 and Y2 are independent.

Answer 1

Solution Suppose u, be independent uniform random vaniables on Co, i Unif (0,1) i:),2 iP. let 7,og (1-U,) X2= (a) then thuir join pd.f indapenclunt Sinie is 1 7 n7017'n7o Con'n) O therwi'se Now e 1-U, Ie Simi ly U2 1-e

Jacobian of tranoformatlon No0 IX- e .(-1 Jl = ax2 -X2 - e +(-1) (x+2) JI Now y def o Jacobian ransformahon - Cx1+ X2 ) X but know that tu Henu f0 fon(2 Where o O. joint dunsity is poduct of mangin al densities ravl Hence X, and independentely dlistributedt X Cme

b) rnd joint density of T/x2 We have and poved ii.d having pobabliry that ame dishibuhon O 0 Since X2 (1 ) Now T2 Jawbian o transformation Now Y2 YI axi 1+ Y2 - Y, I+Y2 YI J 40

formation Jacobian trans Ct') - Y, Y, e -1 whene 0.u O 1-1 Tn Gamana with (shape- 2, Jcale- ) paramutens h peta second kind with C1,) parrametres that Here seen wi podct ot monginals indupendent. Henu T,Ya 1 1 12t i,)10, y220