Question

2. From Kay 8.6, for the PMF PX.y i,1 J O j-1 iO i- 1 (a) Find pyixUld (b) Find pxy lil (c) Find Eyix [Y r] and Vary xY r (d) Find Exy X|y] and Varxy X |yj) (e) Find Ex[X] and Ey Y] from the above

Answer 1

® Solution :- Guiven that JPxy [i, j] /uzo ja list 43 46 Marginal of Px (x=i) = Es Pkiy (e. ) Px(X=0) = I Px.y 10.3) = Px.y 10.0) + Px.y(0.1) = 16+½ = 316 J=0,1 ورنال = 1/2 Px(X=1) = PX,Y (1,1) = Px,y (1,0) + Pxy (11) = 1/3+1/6 = 376 = 42

Marginal of y Py 14 = j) = 7 Px.y(ij) Pyl Y=0) = { Px.yle.o) = Px,y (0,0) + Px.y(0,0) = 46 + 143 1=0,1 = / Py (4=1)= E Px.yli,1) = Px.y (0,1)+Px,y (1,1) = 1/+% = 376 =42.

@ PXIX sulij = P(X=j,x=1) P(x-i) Pyx TY=/x+i)= PIY= 3,X=1) P(x=1) P(Y=)/ X=0) = Ply=j »X=0) P(X=0) P( Y=0/X=0) = P( Y=0.x=0) Plx=0) PLY=0/X=0) = 431 P(Y=1/X=0) = P(Y=1, X=0) P(x=0) 1 PCY=1/x-o) = 2/3) PlY=0/4+1) = P1 Y=0 x=1) P(X=1) | Ply0/x-1) =

@ PlY=1/X=1) = P(Y=1,X=1) P(X=1) | P[Y=1/X=1) = 131 B Pxly lilj) = P1X=i, y=j) Plyj) PLX = il Y=j)= P(X=i,Y*)) PlY-)) P[X=0/Y=0) = P(X=0, Y=0) PCY=0) | PIX=0lY=0) = 430 Plx=1/Y=0) = P(X=1, Y=0) Ply=0) 1 P(X=ply=0) = 43 P(X-01 Y-1) = P(X=0, Y=1) - Ply=1) [Pixotras

© P(X=11Y=1) = P(X=L, Y PCY=1) [P(X=1141) = 43] © EYIX [Y/x] = [y P (Y=j /X-i) goool = oply=0&X=0) + I Ply=1/X=0) 20.4+% EXLXlYX) = ₂1 Eynx 191x) = Ey Ply=j [X= 1) FO PIY=0 /X-1) + I PIY=1/X=1) = 0. + ½ TEXIX {Y 12-1) = 12 51 y el Vy IX 19/-o] = 6,872/=0] + [6418 [YlX=0|2 = % + 479 VYEX [YEx=0] = B14 Answer 37 Eyıx [Ylx=0] = E yoxlPpre

VyLX TY1wa] = Eyex [440=1] + [Ey ixfY[X=11] = '3+ Ya 2.. 23. EylxlY4x=1] = EYLX [Ylx=] Vy ex[Y/x=1] = 244 Answer @ Exty [X 17:0] = [X P(x=1, y=0) 2-0,1 EO PIX=01Y=0) +1.P (x = 1/4 =0) = 0.4+ 1.% | Exly (X14-0] = ² Answer Exty (X1Y=1]= I x P(X= i ly=1) = 0.P(X=0fY=1) + 1: P( X=0/4=1) = 0.2/3 + 4/3 Exty (X1Y=1] = 431 ranswer Valy (X + b)= Ex17 (x1Y-Đ) + ExLY [X Y-” = 4/3 + 449 37 EXlY1X?IY=13= 5x1y[814] Nxly (X1Y=0) = 49 | Anslosest:

Vily ( X 1Y=1) - Exly ( X?/Y=1) = /-/a I vxly (x+y=1) = /9 Answer © Ex[x] = I x PCX=i) X=0,1 E O P(x=0) + 1 PlX=1) = 0./+/ [Ex[x]= 4 Answer Ey (Y) = E y Ply=j) =OPCY=0)+1 PlY=1) = 0.4/2 + 1x 12 = 1/2 Answer | Eyri