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14.1 Problems on conditional expectation, regression

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For the distributions in Exercises 1-3

  1. Determine the regression curve of Y on X and compare with the regression line of Y on X .
  2. For the function Z = g ( X , Y ) indicated in each case, determine the regression curve of Z on X .

(See Exercise 17 from "Problems on Mathematical Expectation"). The pair { X , Y } has the joint distribution (in file npr08_07.m ):

P ( X = t , Y = u )
t = -3.1 -0.5 1.2 2.4 3.7 4.9
u = 7.5 0.0090 0.0396 0.0594 0.0216 0.0440 0.0203
4.1 0.0495 0 0.1089 0.0528 0.0363 0.0231
-2.0 0.0405 0.1320 0.0891 0.0324 0.0297 0.0189
-3.8 0.0510 0.0484 0.0726 0.0132 0 0.0077

The regression line of Y on X is u = 0 . 5275 t + 0 . 6924 .

Z = X 2 Y + | X + Y |

The regression line of Y on X is u = 0 . 5275 t + 0 . 6924 .

npr08_07 Data are in X, Y, P jcalc- - - - - - - - - - - EYx = sum(u.*P)./sum(P);disp([X;EYx]')-3.1000 -0.0290-0.5000 -0.6860 1.2000 1.32702.4000 2.1960 3.7000 3.81304.9000 2.5700 G = t.^2.*u + abs(t+u);EZx = sum(G.*P)./sum(P); disp([X;EZx]') -3.1000 4.0383-0.5000 3.5345 1.2000 6.01392.4000 17.5530 3.7000 59.71304.9000 69.1757
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(See Exercise 18 from "Problems on Mathematical Expectation"). The pair { X , Y } has the joint distribution (in file npr08_08.m ):

P ( X = t , Y = u )
t = 1 3 5 7 9 11 13 15 17 19
u = 12 0.0156 0.0191 0.0081 0.0035 0.0091 0.0070 0.0098 0.0056 0.0091 0.0049
10 0.0064 0.0204 0.0108 0.0040 0.0054 0.0080 0.0112 0.0064 0.0104 0.0056
9 0.0196 0.0256 0.0126 0.0060 0.0156 0.0120 0.0168 0.0096 0.0056 0.0084
5 0.0112 0.0182 0.0108 0.0070 0.0182 0.0140 0.0196 0.0012 0.0182 0.0038
3 0.0060 0.0260 0.0162 0.0050 0.0160 0.0200 0.0280 0.0060 0.0160 0.0040
-1 0.0096 0.0056 0.0072 0.0060 0.0256 0.0120 0.0268 0.0096 0.0256 0.0084
-3 0.0044 0.0134 0.0180 0.0140 0.0234 0.0180 0.0252 0.0244 0.0234 0.0126
-5 0.0072 0.0017 0.0063 0.0045 0.0167 0.0090 0.0026 0.0172 0.0217 0.0223

The regression line of Y on X is u = - 0 . 2584 t + 5 . 6110 .

Z = I Q ( X , Y ) X ( Y - 4 ) + I Q c ( X , Y ) X Y 2 Q = { ( t , u ) : u ≤ t }

The regression line of Y on X is u = - 0 . 2584 t + 5 . 6110 .

npr08_08 Data are in X, Y, P jcalc- - - - - - - - - - - - EYx = sum(u.*P)./sum(P);disp([X;EYx]')1.0000 5.5350 3.0000 5.98695.0000 3.6500 7.0000 2.31009.0000 2.0254 11.0000 2.910013.0000 3.1957 15.0000 0.910017.0000 1.5254 19.0000 0.9100M = u<=t; G = (u-4).*sqrt(t).*M + t.*u.^2.*(1-M);EZx = sum(G.*P)./sum(P); disp([X;EZx]') 1.0000 58.30503.0000 166.7269 5.0000 175.93227.0000 185.7896 9.0000 119.753111.0000 105.4076 13.0000 -2.899915.0000 -11.9675 17.0000 -10.203119.0000 -13.4690
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(See Exercise 19 from "Problems on Mathematical Expectation"). Data were kept on the effect of training time on the time to perform a job on a production line. X is the amount of training, in hours, and Y is the time to perform the task, in minutes. The data are as follows (in file npr08_09.m ):

P ( X = t , Y = u )
t = 1 1.5 2 2.5 3
u = 5 0.039 0.011 0.005 0.001 0.001
4 0.065 0.070 0.050 0.015 0.010
3 0.031 0.061 0.137 0.051 0.033
2 0.012 0.049 0.163 0.058 0.039
1 0.003 0.009 0.045 0.025 0.017

The regression line of Y on X is u = - 0 . 7793 t + 4 . 3051 .

Z = ( Y - 2 . 8 ) / X

The regression line of Y on X is u = - 0 . 7793 t + 4 . 3051 .

npr08_09 Data are in X, Y, P jcalc- - - - - - - - - - - - EYx = sum(u.*P)./sum(P);disp([X;EYx]')1.0000 3.8333 1.5000 3.12502.0000 2.5175 2.5000 2.39333.0000 2.3900 G = (u - 2.8)./t;EZx = sum(G.*P)./sum(P); disp([X;EZx]') 1.0000 1.03331.5000 0.2167 2.0000 -0.14122.5000 -0.1627 3.0000 -0.1367
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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