Question

7. (10 points) Consider two jars, Jar M and Jar W. In Jar M, there are 3 balls numbered 0, 1, 2. In Jar W there are 3 balls numbered 1, 2, 3. A ball is drawn from Jar M, then a ball is drawn from Jar W. Define M as the number on the ball from Jar A and W the number on the ball drawn from Jar B. Let X = MWM times W) and Y = max{W, M}(Maximum of W and M). (a) Find the joint probability mass function of X and Y. (b) Are X and Y independent? Use the definition of independence to support your answer. (c) Compute the Corr(X,Y) (ie, Correlation coefficient of X and Y).

Answer 1

(7) from the observed data The prombility Mass function of Mid me D, 1,2 PM (m) 13 O otherwise PMF of wis Pwlw) X can in [142,33 y=1,2) .P Va WA 1,2,3 otherwise (ai X= MW Ś y s max (Wim) Lake values in 50, 1, 2, 3, 4, 6} y can take values biedt find the P.NE of y ply=y) plyzy) - ply ey-1) pl maxim, w) sy) - pl mal N.W£y.) P[mey, weyl ap(Megal; we you) plmeyj apl wey)-p(Megal) P(weya) (y=1,2) { y + -y+1) = 24 And for y=3 only w can attain value 's .: ply=3) & P | W=3, Mim) pl was) È pluomi P(was). % pl Y=y) 3 otherwise y +1 * solim (4-1) 3 3 y . a 2 mo MED 990 if y = 1,2 iby c 0

Joint PMF FOY X:0 p(x:o, yoy) PlM:0, woy) Ya, y 1,43 For Plxalı yay)Plmal, w=1,yey) p(x=144) I otherwise For x=2 plx=2 yay J - PC M32, , Wal, y= y) +pl Mal, Wally yay) Pl M=2 wal 1) +PIM=1, W=2) ib y=1 Selma PlMel, waidava - foto Iy=2 Othel uise P ( x=2 y=y) (% y = 2 else D For x=3 p(x=3, y=y) Plmol, Was , Y:4) y :3 flori For x=4 plx: 4, Y=y) pl Mz2; w=2, y=y) { y = 2 else FOY x = 6 p( x= 6, y=y) PlX: 6, yay ) pl M=2, . Was, yay ) y=3 sra else

Joint PMF of X & Y ld ol- /9, x=b O 1 plX=Y=y) - 2= 0, y = 1,2,3 Yo! 2=1 Ç yal % 2=2iye ya x=3 ya3 Ya R24 y=2 y = 3 else (6) Marginal PMA of x is plx=0) Ź p(x=0; y=y) y=1 plx=1) plxal, y=1) ala Pl*:2) plx:2 14 = 2 ) % plx3). p(x=3, Y=3) - Ya plx=4) plx:4, 7:2) p(x=6) p(x=614-3) -1° . plxs) 'G Ya id: 1, 3, 4,6 g. 12-0 49 X-2 else 0 Now are X&Y not independent Plx=t& , yay) # Pl x= d ). Plya y.) vay p(xo, Yal) - tai plxo) s ply=1) 3 ** x 9 y are not independenta Hence

(c) cou (xiv) Exy) - 5x). Ely y. Ply-y) a'lo +244 +3*} Ely) ma Y = 1 Elys 1 + 10 19 110 Now Elx) Ith & 24 2 / 2 + 3 + 16 + 4+ 16 + 6*16 ws It 4+3 +4+6 JE(x)=2 Elxy) Exy plxase , Y =y) Iwleta +242x + 343**+ 44274+63** I+P+9+8+18 9 44 9 Elxy) 44 9 ::cou (x,y) - 44 9 * 2 9 44-38 9 : coulx,y) = wory ( x,y) Lou (x,y) NOW, v varx. Nay Narly) 2 3 30 ra 513 5 * 56 cort (x,xla va 553 please give up - vote for my hard work