We now extend the definition of probability distribution from discrete (see Discrete Probability Distributions) 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

Click here for additional information about continuous probability distributions which relies on calculus.

We will study a number of continuous distributions in this website such as the normal distribution and *t* distribution.

I FOUND IT USEFUL

comprehensive summary of the concept of continuos probability,favourable even unto the ‘slow learners’.Well done,bravoo!