Question

Define the joint pmf of (X, Y) by

f(0, 10) = f(0, 20) = 1 / 24, f(1, 10) = f(1, 30) = 1 / 24,

f(1, 20) = 6 / 24, f(2, 30) = 14 / 24

Find the value of the following. Give your answer to three decimal places.

a) E(Y | X = 0) =

b) E(Y | X = 1) =

c) E(Y | X = 2) =

d) E(Y) =

Answer 1

solution

(a) E(Y/X=0)

P(X=0)= Pxy(x=0,y=10)+ Pxy(x=0,y=20)=1/24+1/24 = 1/12

E(Y/X=0)=(Y=10)* {P(X=0,Y=10)} / P(X=0)+(Y=20)*P(X=0,Y=20)} / P(X=0)

E(Y/X=0)=10*(1/24)/(1/12) + 20*(1/24)/(1/12) = 15.000

(b)E(Y/X=1)

P(X=1)= Pxy(x=1,y=10)+ Pxy(x=1,y=20)+ Pxy(x=1,y=30)=1/24+6/24+1/24 = 1/3

E(Y/X=1)=(Y=10)* {P(X=1,Y=10)}/P(X=1)+(Y=20)*P(X=1,Y=20)}/P(X=1)+(Y=30)*P(X=1,Y=30)}/P(X=1)

E(Y/X=1)=10*(1/24)/(1/3)+20*(6/24)/(1/3)+30*(1/24)/(1/3) = (10/8) +15+ (90/24)=1.25+15+3.75 =20.000

(c)E(Y/X=2)

P(X=2)= Pxy(x=2,y=30)=14/24

E(Y/X=2)=(Y=30)* {P(X=2,Y=30)}/P(X=2) = 30*(14/24) / (14/24) =30.000

(d)

E(Y)=(Y=10)*P(Y=10) + (Y=20)*P(Y=20) + (Y=30)*P(Y=30) ....(1)

P(Y=10) =  Pxy(x=0,y=10)+ Pxy(x=1,y=10) = 1/24+1/24 = 2/24

P(Y=20) =  Pxy(x=0,y=20)+ Pxy(x=1,y=20) = 1/24+6/24 = 7/24

P(Y=30) =  Pxy(x=1,y=30) + Pxy(x=2,y=30) = 1/24 + 14/24 = 15/24

from (1)

E(Y)=10*2/24 + 20*7/24 + 30*15/24 = (20+140+450)/24 = 25.417

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