# When does a discrete uniform distribution become a continuous uniform distribution?

I was looking through a question posted that was asking for examples on real-life continuous uniform distributions, and an example given was that of a top, with the degree of rotation divided by 2$$\pi$$ to fulfill a continuous uniform distribution from [0,1].

I am a little confused by this statement. Assuming that we allow the range of angle from [1, 360], wouldn't each angle just have a probability of $$\frac{1}{360}$$? Hence, wouldn't this make it into a discrete uniform distribution?

In other words, what I am trying to ask is, when does a discrete uniform distribution become a continuous uniform distribution? Also, if the example on the top is not a continuous uniform distribution, how can I reconcile this? Is it by allowing the degree of rotation to take on decimal values as well? Can anyone also give more examples of discrete vs continuous uniform distributions?

## 1 Answer

The top example is a continuous uniform distribution – it may assume any real number of degrees, like 7.5° or 123.456789°. The statement that "each angle has a probability of $$\frac1{360}$$ is a discretisation of the original distribution, creating a countable number of bins from the original uncountable (real-valued) sample space.

In general, the difference between discrete and continuous distributions is that a discrete distribution's sample space is countable (finite or in bijection with $$\mathbb N$$), whereas a continuous distribution's is uncountable (in bijection with $$\mathbb R$$). As an example, the number of steps a fixed person takes in covering 100 metres follows a discrete distribution, whereas the average speed in that same distance follows a continuous distribution.

• Got it...thanks! – statsguy21 Oct 8 '18 at 16:17