# Uncertainty Wednesday: Continuous Random Variables

By [Continuations](https://continuations.com) · 2017-07-12

uncertainty wednesday, random variable

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So far in [Uncertainty Wednesdays](http://continuations.com/tagged/uncertainty-wednesday) we have only dealt with models and [random variables](http://continuations.com/post/162082453895/uncertainty-wednesday-random-variables) that had a discrete [probability distribution](http://continuations.com/post/161019122115/uncertainty-wednesday-probability-distribution). Often in fact we had only [two possible states or signal values](http://continuations.com/post/156037078960/uncertainty-wednesday-states-and-signals). There are lots of real world problems though in which the variable of interest can take on a great many values. For example the time between two events taking place. We could try to break this down into discrete small intervals (say seconds) and have a probability per second. Or we could define a [continuous random variable](https://en.wikipedia.org/wiki/Random_variable#Continuous_random_variable) where the wait time can be any real number from some continuous range.  

Now if you have been following along with this series you will have one immediate objection: how can we assign a probability to our random variable taking on a specific real number from a range? A range of reals contains uncountably infinitely many real numbers and hence the probability for any single real value must be, well, infinitely small? So how do we define a Prob(X = x)?

Before I get to the answer let me interject a bit of philosophy. There is a fundamental question about the meaning of real numbers: are they actually real, as in, do they exist? OK, so this is a flippant way of asking the question. Here is a more precise way. Is physical reality continuous or quantized? If it is quantized, then using a model with real numbers is always an approximation of reality. My reading of physics is that we don’t really know the answer. A lot of phenomena are quantized but then there is something like time, which we understand extremely poorly (which is why I chose time as opposed to say distance as my example above). Personally, while not, ahem certain, I am more inclined to see real numbers as a mathematical ideal, which approximates a quantized reality.

Does this matter? Well, it does because too often continuous random variables are treated as some kind of ground truth, instead of an approximation to a physical process. And as we will see in some future Uncertainty Wednesday, often this is a rather restrictive approximation.

Now back to the question at hand. How do we define a probability for a continuous random variable? The answer is through a so-called probability density function (PDF). I find it easiest to think of the PDF as specifying the probability “mass” for an infinitesimal interval around a specific value. Let’s call our density function f(x), then the value of f(x) at x is not the probability of X = x but rather the probability of x - ε ≤ X ≤ x + ε for an infinitesimal ε (I will surely get grief from someone for this abuse of notation).

But by thinking about it this way it then follows quite readily that we can find the Probability of X being in a range by forming the integral of the probability density function for that range

![](https://storage.googleapis.com/papyrus_images/9a51e26caf3d6033a712cea134d2bc2b.png)

Probably the single best known probability density function is the one that gives us a random variable with a [Normal Distribution](https://en.wikipedia.org/wiki/Normal_distribution). The shape of the PDF is why the Normal Distribution is also often referred to as the “Bell Curve”

![image](https://storage.googleapis.com/papyrus_images/08096f3deadb33f729683600dc61cf16.png)

Next Uncertainty Wednesday we will dig a bit deeper into continuous random variables by comparing them to what we have learned about discrete ones.

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*Originally published on [Continuations](https://continuations.com/uncertainty-wednesday-continuous-random-variables)*