Question

(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 = MW(M 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

Solcetion: Griven data: 7 probability mouse function of m is - ) Pm (m) m = 0, 1, 2 0 otherwise PMF Wis - PW (w) / W = 1, 2, 3 otherwise X MW fiya mosi (w m) x can take values in fo, 1, 2, 3, 4, 6} & y can take values in {1, 2, 3} . we first find the p. m.f of Y p(Y=y) =P(Y=Y) -P (YE4-) د را در /

Therwise y3 F if y=1,2 Chaldi 24 aids Tre=m3d 4w Th= wld (s=0 2 Q Cu=WE=m) dz p (y=3) por y=3, only w can attain value 3 And 6 Fire -[ith-ith ] 3 2l=hi (--) ya - hy the (1-65 M) & (1769. M) d-Cham) d (h sui)& (1-6 EM 1-haw) d-Sham hoa ho 5W) de ri-h7 cm'w) Home) d-rhacm'w) kW) d 2

Joint PME 3 for 9- p (x=0, Y>9. p (m=0, w=y) zla Y=1, 2, 3 for x=1 p (x=1, Y- y) = p (m>!, W= 1, Y=y :- P(xal, yoy) ($ (m=1, W=1)=-4, it y=1 , otherwise Por x=2, P (x=2, Y> y) = p (məz, W-l, ysy) 4P (m=1, W=2, Y=y) p (m=2, W=1) +P (m=1, W=2) 0 other wise > y=2 p(x=2, Y=y) 219 echi 0 else

+ for x>3 p(x3,=) > p (m=1, W=3, Y-4) sua 9>3 0 else, hou let p (X=4, Yoy) = P(m=2, w=2, y=y) y=2 is on else for X-6 p (xob, Y=y) =P (m=2, W=3, Y=y) y=3 rya p (xəb, Y=y) else tuis, toond pmf of si a puox - Chak xax) +

5. Þ (x>x, yoy) 9= 0 of y = 1, 2, 3 n=1 a yol You 2/9 19 x=2 Y=2 >3, y=3 yai a = 4, y=2 ya 2-6, y>3. 6 else 3 S b) marginal PMf of X is - P (x=0) > [(X>0,4=y) у? ) p (x=1) p (x=2) P (xo), y=1) > p (x=2,8=2) = 214 p (x= 3, y> 3) = la P (X=3) P (x=4) P (x=4, Y=2) ya ♡ (X=6) p (x=b, y=3) 19 x=0 . P (x>x) Y ya , x=1, 3,4,6 219 / > else,

9 NOW x 9y arenet Independent = (R <(x <x) d P(x>A py) x * P (x> by y l) lm Pxzb) < P (Y D > 지름 4월 Hance x9y are not in dupen deuf. cor (x,y) (x) - Ex) ECG) E () - 4 ) + 144 + 37 y=1 EY) 16 + 의어 119 9 Nobo, E() > [ * + 24 f3+ J +4r +17 + 캐t3H+6 5 E(x) = ' 2. (xy) = Exye (x> X,Y> 1) 개 - 11 fzy{ + 3421

으 b 36 ร-รัฐ 4 - - : Var (x) - 4 (tl +9 tb + 3 - 24 (부 우지 못하 1우길 우수 ) (ECx) (X크 vay (x) Jvar ty) va ly) (AK) 9) 시씨 M) 'non 름 으 FA cor (x,y) . b 82 - Ah- - A5 (4) A 윴 9 (x) 1 b 3] + 3 + 5+3 +1 t 우한 이 + 12.11

8 NOW, var (Y) > FCyY) - (E(Y) 14 + 4 + 34 | Z 2t 쀼 43 9 i var (Y) 25 Y Cory V var (x) vas (4) 를 를 r35 Cord (x,y) NE 뜷 prease give me thumb up