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?
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.