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# HW13 - Homework 13 6.1.1 a Bin(3,0.5 0 1/8 0.40 0.35 0.30...

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Homework 13 6.1.1 a) Bin(3,0.5) 0 1 2 3 1/8 3/8 3/8 1/8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0 1 2 3 b) Bin(3-x, 0.5) Y X=0 X=1 X=2 X=3 0 1/8 - - - 1 3/8 ¼ - - 2 3/8 ½ ½ - 3 1/8 ¼ ½ 1 X=0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0 1 2 3 X = 1 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0 1 2 3 X =2 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0 1 2 3 X=3 0.00 0.20 0.40 0.60 0.80 1.00 1.20 0 1 2 3 c) Y X=0 X=1 X=2 X=3 0 1/64 - - - 1 3/64 6/64 - - 2 3/64 12/64 12/64 - 3 1/64 6/64 12/64 8/64 d) Y Prob 0 1/64 1 9/64 2 27/64 3 27/64

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0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0 1 2 3 e) Y X=0 X=1 X=2 X=3 0 1 - - - 1 1/3 2/3 - - 2 3/27 12/27 12/27 - 3 1/27 6/27 12/27 8/27 Y=0 0 0. 2 0. 4 0. 6 0. 8 1 1. 2 0 1 2 3 Y =1 0. 00 0. 10 0. 20 0. 30 0. 40 0. 50 0. 60 0. 70 0 1 2 3 Y =2 0 0. 1 0. 2 0. 3 0. 4 0. 5 3 0 0. 05 0. 1 0. 15 0. 2 0. 25 0. 3 0. 35 0. 4 0. 45 0. 5 0 1 2 3 f) Y Best guess for x (=highest probability) 0 0 1 1 2 1 or 2 3 2 g) Y Probability of being correct 0 1*1/64 1 2/3*9/64 2 12/27*27/64 3 12/27*27/64
Sum of all the rows=31/64= 0.484375 6.1.2 T G 0 1 2 3 4 Distribution of G 0 1*.1=1/10 ½*.2=1/10 ¼*.4=1/10 1/8*.2=1/40 1/16*.1=1/160 53/160 1 - ½*.2=1/10 ½*.4=2/10 3/8*.2=3/40 4/16*.1=4/160 64/160 2 - - ¼*.4=1/10 3/8*.2=3/40 6/16*.1=6/160 34/160 3 - - - 1/8*.2=1/40 4/16*.1=4/160 8/160 4 - - - - 1/16*.1=1/160 1/160 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0 1 2 3 4 6.2.3 a) P(Y=y|X=x)=1/(n-x) 1<=x<n E(Y|X=x)= 2 1 2 1 ) 2 ) 1 ( * ) ( ) ( * ( 1 1 ) 1 ( * 1 1 + + = + - + = + - - + - - = - = - + = + = x n x n x x n x n x n x x n y x n x n y n x y n x y b) P(X=x|Y=y)=1/(y-1) 2<=y<=n E(X|Y=y)= 1 1 1 * / 2 1 y x x y y - = = - 6.2.4a,d a) E(Y|X=x)=

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