## Two Variables

Often, we are concerned with the joint behavior of two random variables. We may describe such behavior in terms of a joint distribution (density function) such as p(x, y). As suggested before, if such a distribution is partitionable, as p(x, y) = g(X)h(y) (11.18)

then the two variables x and y are independent.

Example. We may develop a simple joint distribution when tossing two dice, where each die represents a random-variable generator. The probability of obtaining any given pair of numbers, say, 3 and 5 [a 3 on the ''X' die and a 5 on the "Y" die is P(X, Y) = P(3, 5)]. For any given pair, of which there are 36 possibilities, the probability is clearly 1/36. Therefore, this is a uniform discrete joint distribution that has only one value, namely, P(X, Y) = 1 /36.

There is also a mean or expected-value concept when dealing with a joint distribution. This may be expressed as

As might be expected, if x and y are independent, the preceding yields the product of the expected, or mean, values of the individual distributions.

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