The Density Function Help

A Pattern Emerges

When working with large populations, and especially with continuous random variables, probabilities are defined differently than they are with small populations and discrete random variables. As the number of possible values of a random variable becomes larger and ''approaches infinity,'' it's easier to think of the probability of an outcome within a range of values, rather than the probability of an outcome for a single value.

Imagine that some medical researchers want to find out how people's blood pressure levels compare. At first, a few dozen people are selected at random from the human population, and the numbers of people having each of 10 specific systolic pressure readings are plotted (Fig. 3-7A). The systolic pressure, which is the higher of the two numbers you get when you take your blood pressure, is the random variable. (In this example, exact numbers aren't shown either for blood pressure or for the number of people. This helps us keep in mind that this entire scenario is make-believe.)

There seems to be a pattern in Fig. 3-7A. This does not come as a surprise to our group of medical research scientists. They expect most people to have ''middling'' blood pressure, fewer people to have moderately low or high pressure, and only a small number of people to have extremely low or high blood pressure.

In the next phase of the experiment, hundreds of people are tested. Instead of only 10 different pressure levels, 20 discrete readings are specified for the random variable (Fig. 3-7B). A pattern is obvious. Confident that they're onto something significant, the researchers test thousands of people and plot the results at 40 different blood pressure levels. The resulting plot of frequency (number of people) versus the value of the random variable (blood pressure) shows that there is a highly defined pattern. The arrangement of points in Fig. 3-7C is so orderly that the researchers are confident that repeating the experiment with the same number of subjects (but not the same people) will produce exactly the same pattern.

Expressing the Pattern

Based on the data in Fig. 3-7C, the researchers can use curve fitting to derive a general rule for the way blood pressure is distributed. This is shown in Fig. 3-7D. A smooth curve like this is called a probability density function, or simply a density function. It no longer represents the blood pressure levels of individuals, but only an expression of how blood pressure varies among the human population. On the vertical axis, instead of the number of people, the function value, f(x), is portrayed. Does this remind you of a Cheshire cat that gradually dissolves away until only its smile remains?

As the number of possible values of the random variable increases without limit, the point-by-point plot blurs into a density function, which we call f(x). The blood pressure of any particular subject vanishes into insignificance. Instead, the researchers become concerned with the probability that any randomly chosen subject's blood pressure will fall within a given range of values.