Question

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 “ maxtW, Mu(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 CorrpX, Y q (ie, Correlation coefficient of X and Y). (d) Find the probability mass function of X given Y “ 3 (ie, ppx|y “ 3q)

Answer 1

Soludon : Probability mass function of м's m0, 1, 2 Pin (m) { 0 otheowise is of w w 0,113 PMF Pw (W) 0 otherwise a x 1 x can Mw & Y - max (Wim) bake values in {0, 1, 2, 3, 4, 63 values in take {1,2,37 & y can of pot > first find the we Plysy ) = Plysy) -Pl YSY-)) iy - 12 = P(max(m,w) sy ) - Plmax (mw ) <y-)) EP (Msy , way) - P (Meyal, w syar) .P(mcy) Plwcy)- P(m cy-lap (way.) 9+! yol Y 1, 2 . Tym 3 3 3 I 14+1-y+)). only w Can attain And for y = 3 value 3. E plu :3 ,H=m) Per=3) mo 2 & Plmam) - PlW=3) FP (W=3). (1)

P(x:2, Yzy ) - PIM 22, Wz1, Vey) ply=y) 2 it y 31, 2 13 if y - 3 o otherwise Joind PMF: for P(x=0, x=y) = P(=0, way) 1,2,3 92 for a=1, plx =1, V=y) - P(1=1, W=1, V-y) it y=1 P(X= 1,8= y) = SPM=1 , = 1) = -2 Promise) for x=2, +P/M = 1, W22, V=y) ع (M = 2,w=1)+P(1921,w=2), it y>2 otherwise O 2% Plx=2, V=y) - {i .y=2 otherwise For 1:3, (وء لار 3 دار (= PM P[x 3, Vry) y=3 > { 0 otherwise For 2034,

plx=4, Yay) - P(M=2,W=2, V=2) n y = 2 {": O Otherwise x 36, For PIXE6, Y=y) = P(M=2,W=3, Yey) p(x=6,x=y) 4:3 {": Ow Thus, joint pmf of x and y is PlX", Yay) el- x=0, & y = 1,2,3 Va , 0421 dy1 219 119 n = 3 2 Ya 174=2,8 y 22 &y: 3 x = 4 dy 126 d y = 3 otherwise 6 3 9 3 P (x=0) /9 Marginal pot of x is 3 E PIX20, Y-y) Y 1 P(x-1) Plx-1, Y =1) P/x-2) p Ix=2, V=2) = 2/9 P/x3) P (x23, Y-3) plx4) = PIX 34, 4=2 ): 119 ܗܐ ܀

DIX = 6 ): p(x26,Y=3 ) : 19 Plx:1) = 113 / 119 x-1,3,4,6 119 , 2 - 2 6 else Nolo are not in dependent P(x), Vry) # P (**») PIV=Y) *xy Here, plX20, V-1) = 1 9 P(x=0).P (V=)) - Ž X ola # 16 Hence X and y cre independent E (XV) - E (x) E (Y) Cov(x, y) 3 tevis Ey.P(V=Y) = 1xş+2x+3xş yal - 19 9 E (*): +2x +3 ** ***+ * 1+4+3+ 4 + 6 2. 9 ElXX ) - E dyp (x=2, Yzy) niy 2 x2 x3 + 3x3 x + 4x 3x4 - lxix + 9 9 +6* 3* Elxy) 44 9

(ov XV). 19)/2) 44 - 38 9 - 2 3 • Co-A ( x + ), (ev (x, y) Var (x) Var (y) var (x) E lx?) - (F(x)) ? (1244+2*** *3*** *** *to*x4) 12) -4 9 vas (x)= 30 9 2 var cyla (12) -(E (Y))? -(** +2*** *3*** )- (19) capproximately) 9 - Cor (x,y) Coro (x,y) a vas lx) Vas (y) 2 213 30 x 5 30 x 5 V23 rsrs 513

a plx=7 14-3) - P(x, 1=3) PV =3) P(x=4 , Y-3) A13 Pix:+/y: 3 ) : 3 xta 220, 3,6 fore otherwise 113 ,01-0, 3,6 0 otherwise Please like and support. Thank you