Pairs of discrete random variables

For many problems there are situations involving more than a single random variable. Consider the case of two discrete random variables, x and y. Here y may take any of the values of the set y. Now, instead of having PMFs of a single variable, the joint PMF, P(x, y), is required. This may be viewed as the probability of x taking a particular value and y taking a particular value. This joint PMF must satisfy 

Take a simple example of the weather - whether it rains or not on two particular days. Let X be the random variable associated with it raining on dt 1, y being that for day 2. The joint PMF may be described by the table 

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Consider the situation in which a chemist randomly samples a bin of pharmaceutical granules by taking n aliquots of equal convenient sizes. Chemical analysis is then performed on each aliquot to determine the concentration (percent by weight) of pseudoephedrine hydrochloride. In this example, measurement of concentration is referred to as a continuous random variable as opposed to a discrete random variable. Discrete random variables include counted or enumerated items like the roll of a pair of dice. In chemistry we are interested primarily in the measurement of continuous properties and limit our discussion to continuous random variables. 

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