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idemo1 % Joint matrix in datafile idemo1
P = 0.0091 0.0147 0.0035 0.0049 0.0105 0.0161 0.0112 0.0117 0.0189 0.0045 0.0063 0.0135 0.0207 0.0144
0.0104 0.0168 0.0040 0.0056 0.0120 0.0184 0.0128 0.0169 0.0273 0.0065 0.0091 0.0095 0.0299 0.0208
0.0052 0.0084 0.0020 0.0028 0.0060 0.0092 0.0064 0.0169 0.0273 0.0065 0.0091 0.0195 0.0299 0.0208
0.0104 0.0168 0.0040 0.0056 0.0120 0.0184 0.0128 0.0078 0.0126 0.0030 0.0042 0.0190 0.0138 0.0096
0.0117 0.0189 0.0045 0.0063 0.0135 0.0207 0.0144 0.0091 0.0147 0.0035 0.0049 0.0105 0.0161 0.0112
0.0065 0.0105 0.0025 0.0035 0.0075 0.0115 0.0080 0.0143 0.0231 0.0055 0.0077 0.0165 0.0253 0.0176
itestEnter matrix of joint probabilities P
The pair {X,Y} is NOT independent % Result of test
To see where the product rule fails, call for D
disp(D) % Optional call for D 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0
Next, we consider an example in which the pair is known to be independent.
jdemo3 % call for data in m-file
disp(P) % call to display P 0.0132 0.0198 0.0297 0.0209 0.0264
0.0372 0.0558 0.0837 0.0589 0.0744 0.0516 0.0774 0.1161 0.0817 0.1032
0.0180 0.0270 0.0405 0.0285 0.0360
itestEnter matrix of joint probabilities P
The pair {X,Y} is independent % Result of test The procedure icalc can be extended to deal with an independent class of three random variables. We call the m-procedure icalc3 . The following is a simple example of its use.
X = 0:4;
Y = 1:2:7;Z = 0:3:12;
PX = 0.1*[1 3 2 3 1];
PY = 0.1*[2 2 3 3];
PZ = 0.1*[2 2 1 3 2];
icalc3Enter row matrix of X-values X
Enter row matrix of Y-values YEnter row matrix of Z-values Z
Enter X probabilities PXEnter Y probabilities PY
Enter Z probabilities PZUse array operations on matrices X, Y, Z,
PX, PY, PZ, t, u, v, and PG = 3*t + 2*u - 4*v; % W = 3X + 2Y -4Z
[W,PW] = csort(G,P); % Distribution for W
PG = total((G>0).*P) % P(g(X,Y,Z) > 0)
PG = 0.3370Pg = (W>0)*PW' % P(Z > 0)
Pg = 0.3370 An m-procedure icalc4 to handle an independent class of four variables is also available. Also several variations of the m-function mgsum and the m-function diidsum are used for obtaining distributions for sums of independent random variables. We consider them in various contexts inother units.
In the study of functions of random variables, we show that an approximating simple random variable X s of the type we use is a function of the random variable X which is approximated. Also, we show that if is an independent pair, so is for any reasonable functions g and h . Thus if is an independent pair, so is any pair of approximating simple functions of the type considered. Now it is theoretically possible for the approximating pair to be independent, yet have the approximated pair not independent. But this is highly unlikely . For all practical purposes, we may consider to be independent iff is independent. When in doubt, consider a second pair of approximating simple functions with more subdivision points. This decreases even further the likelihood of a falseindication of independence by the approximating random variables.
Suppose exponential (3) and exponential (2) with
Since , we approximate X for values up to 4 and Y for values up to 6.
tuappr
Enter matrix [a b] of X-range endpoints [0 4]
Enter matrix [c d] of Y-range endpoints [0 6]
Enter number of X approximation points 200Enter number of Y approximation points 300
Enter expression for joint density 6*exp(-(3*t + 2*u))Use array operations on X, Y, PX, PY, t, u, and P
itestEnter matrix of joint probabilities PThe pair {X,Y} is independent The pair has joint density . It is easy enough to determine the marginals in this case. By symmetry, they are the same.
so that which ensures the pair is independent. Consider the solution using tuappr and itest.
tuappr
Enter matrix [a b] of X-range endpoints [0 1]
Enter matrix [c d] of Y-range endpoints [0 1]
Enter number of X approximation points 100Enter number of Y approximation points 100
Enter expression for joint density 4*t.*uUse array operations on X, Y, PX, PY, t, u, and P
itestEnter matrix of joint probabilities P
The pair {X,Y} is independent 
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