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For the distributions in Exercises 1-3
(See Exercise 17 from "Problems on Mathematical Expectation"). The pair has the joint distribution (in file npr08_07.m ):
| 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 .
The regression line of Y on X is .
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 (See Exercise 18 from "Problems on Mathematical Expectation"). The pair has the joint distribution (in file npr08_08.m ):
| 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 .
The regression line of Y on X is .
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 (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 ):
| 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 .
The regression line of Y on X is .
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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