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part28

part28 - -4.9 a For x> 0 y> 0:1[Y 2/2FXY(x y = Jo Jo...

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Unformatted text preview: .--4.9 a) For x > 0, y > 0[:1: [Y 2/2FXY(x, y) = Jo Jo axe-a:1: 2bye-by /2dxdy= (1 - e-ax2/2)(1 - e-by2/2)b) P[X>y] = l°O1xaxe-ax2/2bye-by2/2dydX[00 2 2= Jo axe-a:1: /2(1 - e-b:1: /2)dxa= 1--a+bc)Fx(x) = lim Fx(x,y) = 1- e-a:1:2/2 x> 0y-+oo* fx(x) = 1;Fx(x) = axe-ax2/2 x > 0Similarlyfy (y) = bye-by2 /2100(-1)00 fx(x) = xe-xe-xYdy = xe-:1: -e-XY = e-:1:4.13 0 x f()-100 -:1:(l+Y)d- e-x(l+y) ((1 + y)x - 1)100 y y - xe x -0 (1 + y)2 1-- (1+y)24.17 a) P[X = i,Y ~ y] = P[Y ~ ylX = i]P[X = i]If X = 1:1 1 [y-l aP[Y ~ ylX = l]P[X = 1] = P[N + 1 ~ y]"2 = "2 J-oo 2e-a/zldz{!ea(y-l) y < 1= 1(2 - e-a(y-l)) y > 1X = -1 is obtained in similar fashion:{!ea(y+l) y < -1P[Y ~ ylX = l]P[X = -1] = 1(2 - e-a(y+l)) y ~ -1~b) Fy(y) = pry ~ ylX =]P[X = 1] + pry ~ ylX = -1]P[X = -1]1 1= P[N + 1 ~ y]2 + P[N - 1 ~ y]2d Id Idfy(y) = dYFy(y) = 2dYFN(Y - 1) + 2dYFN(Y + 1)1 1= 2fN(y - 1) + 2fN(y + 1)- "'"'...,.".~_. ///II\~~~-I ;..y///'/"""" :~---//'~"' I I I ~-1 0 1 Yc) P[X = 11Y > 0] = pry > OIX = I]P[X = 1] = 1 - ~e-a...
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part28 - -4.9 a For x> 0 y> 0:1[Y 2/2FXY(x y = Jo Jo...

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