2.10 PROBABILITY DENSITY FUNCTIONSMany random variables encountered in practice are distributed over a continuous rather than a discrete set of values. Examples include the time one waits at a bus stop until the next bus arrives, the tons of trash collected in a city on a given day, the distance a social worker in the field will travel on a given day, and the amount of electricity consumed by a household during a year. Just as probability mass functions (pmf's) allowed us to explore the probabilistic behavior of discrete random variables, probability density functions (pdf's) allow us to do the same for continuously distributed random variables. We define a pdf for the (continuous) random variable X as follows: fx(x) dx ![]() Note that our definition is not stated in terms of the probability that random variable X assumes exactly the value x; for a purely continuous random variable, this probability is zero. Thus, in order to make any probability statement using pdf's, one must integrate the pdf (even if only over an infinitesimal interval of length dx). Some random variables occurring in practice are mixed; that is, they have a purely continuous part and they have a discrete part, An example could be the location of a bus at a random time along a straight-line street route; the bus might be viewed as uniformly distributed over the route except for a probability pi of being located at X = xi, the location of the ith stop (i = 1, 2, . . . , N). In this case ![]() ![]() Since probabilities must be nonnegative, we must have fx(x) ![]() ![]() The cdf grows linearly from zero at U = a to 1 at U = b (Figure 2.8). The uniformly distributed random variable is often implied when the term "random" is used in problem statements, although we will attempt to avoid such ambiguous terminology here. ![]() The compound pdf allows us to study two or more continuous random variables simultaneously. For two random variables X and Y, their compound pdf is given by ![]() In cases involving multiple random variables, X1, X2, . . . ,XN, one may still be interested in the marginal pdf for Xi,fxi(xi), defined so that fxi(xi)dxi ![]() ![]() ![]() ![]() |