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2.4 Random Variables in Applications > 2.4 Random Variables in Applications - Pg. 31

2.4. R A N D O M V A R I A B L E S IN A P P L I C A T I O N S 31 C h e s t X r a y could be positive or negative, and the state of S m o k i n g H i s t o r y could be yes or no, where yes might mean the patient has smoked one or more packs of cigarettes every day during the past 10 years. After distinguishing the possible values of the random variables (i.e., their spaces), we judge the probabilities of the random variables having their values. However, in general, we do not directly determine values in a joint probability distribution of the random variables. Rather, we ascertain probabilities, con- cerning relationships among random variables, which are accessible to us. We can then reason with these variables using Bayes' Theorem to obtain probabil- ities of events of interest. The next example illustrates this. E x a m p l e 2.29 Suppose Sam plans to marry, and to obtain a marriage licence in the state in which he resides one must take the blood test ELISA (enzyme linked immunosorbent assay) which tests for the presence of H I V (human im- munodei~ciency virus). Sam takes the test and it comes back positive for HIV. How likely is it that Sam is infected with HIV? Without knowing the accuracy of the test, Sam really has no way of knowing how probable it is that he is infected with HIV. The data we ordinarily have on such tests are the true posi- tive rate (sensitivity) and the true negative rate (specificity). The true positive rate is the fraction of people who have the infection that test positive. For example, to obtain this number for ELISA 10,000 people who were known to