Continuous Uniform Distribution

I saw a statement on an answer to a question saying:

A continuous uniform distribution is defined by upper & lower limits.

When I was thinking about it I thought there is no limits? Isn’t a contiinous variable one that has no bounds. Eg. like the specific number of mm of rain on a day, as you can count in cm,mm, and smaller etc. Whislt a discrete random variable is one that is an absolute whole value?

In that case would it not be a discrete uniform distribution with set upper and lower bounds?

I’m getting confused.


Think of the numbers between 0 and 1. There are infinite number of decimals that are between the two, but they are bound by 0 and 1

When we say that a distribution is continuous, we simply mean that it can take on any (real number) value between its limits. Your rain example is a good one: it has a lower limit of 0mm and an upper limit that’s probably less than, say, 1,000,000mm; between those limits, it can be any (real) value (you could get 10mm, or 10.1mm, or 10.01mm, or 10.001mm, or . . .).

Similarly, continuous uniform distribution has a lower limit and an upper limit, but between those limits it can take on any (real) value: if its limits are 1 and 5, for example, then it could be 2.5, or 2.51, or 2.50385703 . . ., or . . .).

Bear in mind, however, that the mere fact that there are infinitely many possible outcomes does not, by itself, make a distribution continuous. (I know that you’re not saying that here, but it’s an important point, and many people get it wrong.) A distribution for which the possible outcomes are all of the whole numbers (1, 2, 3, . . .) has an infinite number of possible outcomes, but is discrete, not continuous.

Yes thank you for clearing that up!

Thanks S,

Does that mean that discrete distributions also have upper and lower limits?

My pleasure.