Question

2. Let X and Y be independent, standard normal random variables. Find the joint pdf of U = 2X +Y and V = X-Y. Determine if U and V are independent. Justify.

Answer 1

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Joint Xand Jan Jaint pdh Notw f U_and f..u, joint pdp of x and y in temp 1J1 of U& y -2uu 18 UV - Qu+5U uv) UV co77 check that and To independlent ar e U and Hexe V Qinear X and y , Whoie are Combinatfon Variable of xandom _ndepeslest nexmaly olastuted the liheax combinattonof know that distseibuted distsibution Jale NOTE: andom Variable 13 Carnespending Exbectaticn Alaxmal 4Variance

Noumally distuileuted and Nre Handom Vorriable 2. Etx)t Ety) EtaxFyJ E(x) ECY)- O ECV)=ECX-Y). YoNtort Va (u) Var (2x+y)= 4 Var (x)Vor ty) +4 Cov(Xy) 4+1 +0 - 5 Var V)Var (X-Y) Vor Cx) Yar CYJ -Qoov(x,y) UNNO,5) VN Nlo,2) . If X and Naumally distsubuted X and Y NOTE aLe thdeperdent Varyable then random ане between Covaruance ndabende nt cheekre Bhat to ate Now Sad have to find the covorianee between we CoVardance between U дnd в V are U and V. o(zero) if (ndependlent& equal not indepenodent then betoeen u and CoVoruance V are U and V are о (Zero) to otaxty, X-Y Varx)-coViX,y) Covt X, y2- Vox (Y) Cov(U,VJ

Cov(uN) 2.1--2.0 0-L T T- Cov (UN). Therefore not Thdepondent andom V and Cre Varuable