Now f(X = 1, Y = 1) = 1, f(X = 1) = 1 (obtained by summing the first column), and f(Y = 1) = 1 (obtained by summing the first row). Since f(X, Y) = f(X) f(Y) in this example we can say that the two variables are statistically independent. It can be easily checked that for any other combination of Xand Yvalues given in the above table the joint PDF factors into individual PDFs.
It can be shown that the Xand Yvariables given in Example 4 are not statistically independent since the product of the two marginal PDFs is not equal to the joint PDF. (Note: f(X, Y) = f(X)f(Y) must be true for all combinations of Xand Yif the two variables are to be statistically independent.)
Continuous Joint PDF. The PDF f (x, y) of two continuous variables X and Y is such that f (x, y) > 0
/ f (x, y)dxdy = P(a < x < b, c < y < d) Jc Ja
APPENDIX A: A REVIEW OF SOME STATISTICAL CONCEPTS 877
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