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The pair has the joint distribution (in m-file npr08_06.m ):
Determine whether or not the pair is independent.
npr08_06 Data are in X, Y, P
itestEnter matrix of joint probabilities P
The pair {X,Y} is NOT independentTo see where the product rule fails, call for D
disp(D)0 0 0 1 1
0 0 0 1 11 1 1 1 1
1 1 1 1 1 The pair has the joint distribution (in m-file npr09_02.m ):
Determine whether or not the pair is independent.
npr09_02 Data are in X, Y, P
itestEnter matrix of joint probabilities P
The pair {X,Y} is NOT independentTo see where the product rule fails, call for D
disp(D)0 0 0 0 0
0 1 1 0 00 1 1 0 0
0 0 0 0 0 The pair has the joint distribution (in m-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 |
Determine whether or not the pair is independent.
npr08_07 Data are in X, Y, P
itestEnter matrix of joint probabilities P
The pair {X,Y} is NOT independentTo see where the product rule fails, call for D
disp(D)1 1 1 1 1 1
1 1 1 1 1 11 1 1 1 1 1
1 1 1 1 1 1
For the distributions in Exercises 4-10 below
on the circle with radius one, center at (0,0).
Not independent by the rectangle test.
tuappr
Enter matrix [a b]of X-range endpoints [-1 1]
Enter matrix [c d]of Y-range endpoints [-1 1]
Enter number of X approximation points 100Enter number of Y approximation points 100
Enter expression for joint density (1/pi)*(t.^2 + u.^2<=1)
Use array operations on X, Y, PX, PY, t, u, and Pitest
Enter matrix of joint probabilities PThe pair {X,Y} is NOT independent
To see where the product rule fails, call for D % Not practical-- too large on the square with vertices at (see Exercise 11 from "Problems on Random Vectors and Joint Distributions").
Not independent, by the rectangle test.
tuappr
Enter matrix [a b]of X-range endpoints [0 2]
Enter matrix [c d]of Y-range endpoints [0 2]
Enter number of X approximation points 200Enter number of Y approximation points 200
Enter expression for joint density (1/2)*(u<=min(1+t,3-t)).* ...
(u>=max(1-t,t-1))
Use array operations on X, Y, PX, PY, t, u, and Pitest
Enter matrix of joint probabilities PThe pair {X,Y} is NOT independent
To see where the product rule fails, call for D 
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