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).

(d) Find the probability mass function of X given Y = 3 (ie, p(x|y = 3))

Answer 1

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7. Probabili ility mass function Mis- ction of p (m ) = I 3 m = 0, 1, 2 otherwise 2 PMF of W is b (w) Just W = 1, 2, 3 otherwise Xean take values in -- a] X=MW & Y= mon (w,m). {0, 1, 2, 3, 4,6} & Y can take values in {1, 2, 3 we first pomy of Y. 1 find the

6 hza [ith-ith] g 난 w { h E (1h) yt 2 (1 hs Md (1 hs W) d hi md (ha wld (1.25m 1 hg wld - Chim n hawld (ths (M"W) koun) s - ( 23 (mw) hamd 12 4 z hl (1 - 2 = x 1 d (hs hd = (h=2)

And four y=3 w can Only attain value 3. mzo 2 Pl Y= 3) = {p(W PW-3, M = m) P(W =3) { P (Mam) .P/w=3). mad I 3 '. Ply=y) = 2y g ify = 1,2 -) if 1/3 3 y = 2 otherwise 2

Toint PME. for Nz o P(x=0 Y=y) Pem M=0, W=y) Ho y=1, 2, 3

for xz1 Wz 2 P(X= 1, Y=y) Pl M=1 21, Yay) 1. P(x-2, Yzy) e SP (M2), W =)) = if yol othes wise. 2 for uzz P(x=2, Y=y) = P(M22, W=l, Yay) FP(M=1, W=2, Y=Y). fel M= 2, Wz 1) + P (M= 1, W=2) 29=2. otherwise P(X= 2, toy) = 2 19. y=2 foor azz else PtX = 3,7 =9) PEME!, W = 3, Y=y) 1902 CESSERXENON 1/2 = 3 delse

གས་བ་རྒྱུ་བ། for P ( X = 4, Y=y) = P(M22, W = 2, Y = y) les Lito y = 2 Z O tre else for Rz 6. f(M=2, w 23,4²4) 44 M2ZXR099 frag Y=3 P(X26,Y)) else Thus, Joint pmf of X and Y is no & Y=1,2,3 Az & yel z ول 779 219 1129 119 n=2, 122. N23. Y-3 2 1019 124, y = 2; uz6, 9=3 else.

Marginal pay of x is- P(X= 0) = P(x=0, Y = y) 3 1 3 yzl PCXzl) = P(x2); Y =)) 179 P ( X = 2) P(x-2, Y = 2) = 2/9 f(x - 2) P(x= 3, Y= 3) op. 19 P (X= 4) P(X=4, Y = 2 10g plag PCX36) P(X=6, Y = 3

( - D су (173 11.9 2 1, 2,4,6.. а (9 Иг 2 eise 2

Now X&Y are not independent 1 PLX=X,Y29) + P { x= x) PLY=y) xuy + z 2 P(X=0) a P (Y=1) F # 3 2 9 Hence X & Y are not independent c) cov (44) E(XY) – E(X) ELY) Ely) - ? y. P(x-4) = 1 x 2 + 2xy +39 1 3 yel 3 1 + 그 10 g 19 g Now, Elx) = la I t ia g att uxlt 6x. 9 9 9

(1) + 4 +3++6 го [[x) 2 , ElxY) = LP(X-X, Y-3) 1,5 1к1 « 1 + 2K L - + 343 1 9 2 9 + Чал) + 6 КЗК) 9 9 ( + 2 ++ +} + I2 9 (17) = 44 9). Coul X,Y)= ५५ 19 2 x 2 44 - 3 г. رہی -- (os(x,1) ( on(x, y) - Cov (x,y) Ју»(8) VAZY) Now

2 var(x)= Z 티씨 E() - 11 + +3 1+1] 9 ㅏ6 우 T 22 나니+9 커6436 2 4 9 Var (*) 66 니 30 66-36 9 N005 ㅋ ElY Vany)= EYY 름 ( + 2 - +3 X 9 2 +3 8라6 9 5 Van (Y)

Coros (X, Y) = Coulx, 4) Juan (x) valy) 2 ار N16 Les 30 5 56 z 85 Coror(x,y) = 2 (Ans 55 d P( X Y =3) PLX = n, Y=3 -P(4=3 P(X=X, Y=3) 1/3 ?}{x22|1=3) - falleg =0,3,6. else. o ) 113 2 Z 120, 3,6 else