Question

Let the joint density function of X and Y be given by the following x +y for 0 < x < 1 and 0 < y < 1 f(x, y) = 0 otherwise Find E[X], E[Y], Var[X], Var[Y], Cov(X,Y), and px,y Find E[X]Y], E[E[X|Y]], and Var[X|Y]. Find the moment generating function Mx,y(t1, t2)

Answer 1

To fines the ElX) ELY) Narcx9, varly). Coxlx, y2 and Pixy) first we find the marginal baf of x andy Marginal baf of a ce on Je Fayjay af'(x+y)dly How at OLALI

Marginal belf of Yle fly= 1° f (x.gadze Let'(x+y) dx_ Iftyt = 4+2 obytt, (110 NOLA E(X)= 5 x + (x olx - 5043244 - ***** = + A EN F / Answer Nole, Ely) = f °y flyselyi

=15+] - IL! An2up] © We know thert Vai(x) = : E(X2) - LED NOW E(XY)= (v2fluid - fix for yox , 4 . . = 32 I TEX & Answer Answert lithere fore var (x)= E(X2)- [E(x) 7² libertatea sa o dies in 121 144. = 60-461

nar(x) = - 1441 Answer @ Similarly Warly)= E (42)– (E14)74 Now, Ely2 = 1 y 2 flys ly = y 2/4 + +) ay y=- TELY - Z Therefore varly) = 5 - 49 12 144 to 49 44 Warly 51 I Answer @ We know theatlov (X,Y)= ELXY) – E(XJEV) Now, ELXV= 1° Tº my flwys dydis

y, 0 ) (X+MJ dy els ( 0 , : у 12) dly ax. -() ах — Fx) - | hеле? Фле, Coutx, Y) = 1 - - {2 - " 12 3 144 -t44 Cy(x,y) = = 44 дител - Cov (X,Y) Jan(x) Vanty) - 144 ++++ +++

- x 144 144IL Itthon = - Id Answer 2. To find ElXlY), FIE (X1) f Var(x/4). first we fines the evneitiones belf of f(x4) We know that fixty= 11942 fly) +(X1Y) = (uty) OLALL (y + 2) © EXNE ( 2 f(x19) els Tab+Y) dx Texty = 24+ 244 Answer

® ETE(XXV)] = E(X) = Answer E[E/X/YİBELY) of ELEVIS € 1① we know that, Vorfly) = 5(x241) – JĒ/X17]2 NOW, E1*4*3=4542f(x4) dx tes - fxIxty) da : f (x2+xy) elu 2y + do – 2 1824 + 433) qyti 14310 There are ot Il var(x64)= 2 * y 24t 14 (24+0)? | 3 MartxNY) Ilit 2uth/14 3 -474) 277 (2y 11: 3 2 17 Answer

Moment generating - BunetionMxylti.fr) Mxx tut) = Eletix+tor] : - SI' ptixt tol(xtysely dy -14 Ixty Myda =fo see tox ta' Eye to * t19 Joly the = f'p tarif' nethely + 19 yet they do party luce = Mee- le Pontogenation by a FPJ4e**+(jo tas ir play oly) and ad fetis fuerte etz-z letzyli jak to o t2 do tz fotzetik dx + 1'1. ptix.clx

etz (lxebx dx a etz stotix. xetix tz Jor -- ti do ed etix da tif etix an Il tu? do 120 Itzt a lta te te cletz- laetix 1 - 1 etix ax] + /t2 et2 _ et24 7 ti To do ti I t2² letzet ti Il ti - lett tr T. +/tzet² _etzt!) Itz Tetrx letz-y let - etti. 1 Et ti ti t2 Tt To tzetz_ptzt!) + 2 till باکاندا