We now extend the definition of probability distribution from discrete to continuous random variables. Such variables take on an infinite range of values even in a finite interval (weight of rice, room temperature, etc.).

While for a discrete random variable x, the probability that x assumes a value between a and b (exclusive) is given by

the frequency function f of a continuous random variable can assume an infinite number of values (even in a finite interval) and so we can’t simply sum up the values in the ordinary way. For continuous variables, the equivalent formulation is that the probability that x assumes a value between a and b is given by

i.e. the area under the graph of y = f(x) bounded by the x-axis and the lines x = a and x = b.

Figure 1 – Area under the curve

Definition 1: For a continuous random variable x is a frequency function, also called the probability density function (pdf) provided:

The corresponding (cumulative) distribution function F(x)  is defined by

Property 2: For any continuous random variable x with distribution function F(x)

Observation: f is a valid probability density function provided that f always takes non-negative values and the area between the curve and the x-axis is 1. f is the probability density function for a particular random variable x provided the area of the region indicated in Figure 1 represents the probability that x assumes a value between a and b inclusively. Note that the probability that f takes any particular value a is not f(a). In fact for any specific value a, the probability that x takes the value a is considered to be 0.

Essentially the area under a curve is a way of summing when dealing with an infinite range of values in a continuum. For those of you familiar with calculus