The probability density function
So far, we have considered the cumulative distribution function as the main way to describe a random variable. However, for a large class of important models, the probability density function (pdf) is an important alternative characterization.
To understand the distinction between the cdf and pdf, we need the notion of probability. In the context of random variables, probability simply means the likelihood that the random outcome falls within a certain range of values, normalized to a number between 0 and 1. For example, let's consider the example of women's heights discussed in the previous section. We concluded that 42.8% of women have a height between 63 inches and 68 inches. An alternative way to express this is to say that, for the random variable that represents women's heights, the probability that the outcome is between 63 and 68 is .428.
The main distinction between the cdf and pdf is the way probabilities are represented by each of them:
For a cdf...
