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The pair { X , Y } has the joint distribution (in m-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

Determine the marginal distributions and the corner values for F X Y . Determine P ( 1 X 4 , Y > 4 ) and P ( | X - Y | 2 ) .

npr08_07 Data are in X, Y, P jcalcEnter JOINT PROBABILITIES (as on the plane) P Enter row matrix of VALUES of X XEnter row matrix of VALUES of Y YUse array operations on matrices X, Y, PX, PY, t, u, and P disp([X;PX]') -3.1000 0.1500-0.5000 0.2200 1.2000 0.33002.4000 0.1200 3.7000 0.11004.9000 0.0700 disp([Y;PY]') -3.8000 0.1929-2.0000 0.3426 4.1000 0.27067.5000 0.1939 jddbnEnter joint probability matrix (as on the plane) P To view joint distribution function, call for FXYdisp(FXY) 0.1500 0.3700 0.7000 0.8200 0.9300 1.00000.1410 0.3214 0.5920 0.6904 0.7564 0.8061 0.0915 0.2719 0.4336 0.4792 0.5089 0.53550.0510 0.0994 0.1720 0.1852 0.1852 0.1929 M = (1<=t)&(t<=4)&(u>4); P1 = total(M.*P)P1 = 0.3230 P2 = total((abs(t-u)<=2).*P) P2 = 0.3357
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The pair { X , Y } has the joint distribution (in m-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

Determine the marginal distributions. Determine F X Y ( 10 , 6 ) and P ( X > Y ) .

npr08_08 Data are in X, Y, P jcalc- - - - - - - - - Use array operations on matrices X, Y, PX, PY, t, u, and Pdisp([X;PX]')1.0000 0.0800 3.0000 0.13005.0000 0.0900 7.0000 0.05009.0000 0.1300 11.0000 0.100013.0000 0.1400 15.0000 0.080017.0000 0.1300 19.0000 0.0700disp([Y;PY]')-5.0000 0.1092 -3.0000 0.1768-1.0000 0.1364 3.0000 0.14325.0000 0.1222 9.0000 0.131810.0000 0.0886 12.0000 0.0918F = total(((t<=10)&(u<=6)).*P) F = 0.2982P = total((t>u).*P) P = 0.7390
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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 m-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

Determine the marginal distributions. Determine F X Y ( 2 , 3 ) and P ( Y / X 1 . 25 ) .

npr08_09 Data are in X, Y, P jcalc- - - - - - - - - - - - Use array operations on matrices X, Y, PX, PY, t, u, and Pdisp([X;PX]')1.0000 0.1500 1.5000 0.20002.0000 0.4000 2.5000 0.15003.0000 0.1000 disp([Y;PY]') 1.0000 0.09902.0000 0.3210 3.0000 0.31304.0000 0.2100 5.0000 0.0570F = total(((t<=2)&(u<=3)).*P) F = 0.5100P = total((u./t>=1.25).*P) P = 0.5570
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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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