What is a Continuous Random Variable?

A continuous random variable is one which has a probability density function, or pdf. That is, a function where the area underneath it over a range of values gives the probability that the variable lies within that range.

Typical continuous random variables can take any value throughout an interval. For example, suppose $X$ represents the exact time, in minutes, until the next bus arrives. It might be 2.5 minutes, 2.53 minutes, 2.5317 minutes, and so on.

This is very different from a discrete random variable, whose possible values can be arranged into a list.

For a continuous random variable, the probability of any one exact value is zero:

$$\mathbb{P}(X=x)=0.$$

This can seem strange at first. The bus must eventually arrive at some particular time! But probability is spread across ranges of values rather than concentrated at individual points.

So, instead of asking for the probability that $X$ is exactly 2.5, we might ask for the probability that the bus arrives between 2 and 3 minutes from now:

$$\mathbb{P}(2\leq X\leq 3).$$

For a continuous random variable $X$, we write its probability density function as $f_X(x)$. Probabilities are given by areas under the pdf. In particular,

$$\mathbb{P}(a\leq X\leq b)=\int_a^b f_X(x),dx.$$

The most important example in introductory statistics and econometrics is the normal random variable. As shown on the slide, we can imagine repeatedly “rolling” for a normal random variable $Z$, with each roll producing a different numerical value.

The key point is that the height $f_X(x)$ of the pdf is not itself a probability, and indeed usually not of much interest. It is the area under the curve across a range of values that gives us probabilities.