Notation inconsistency? Standard uniform distribution U(0,1)

Question

Very simple and quick question. Usually distribution notation is such that you give the name of the distribution, then its mean, and finally the variance, for example for normal distribution:

$$N(0,1)$$

The 0 means that the distribution has mean zero, and the 1 tells that the variance is one. However, for standard uniform distribution:

$$U(0,1)$$

The zero is the minimum value the distribution generates and 1 is the maximum value. Using the more standard notation it should be:

$$U(0.5, \frac{1}{12})$$

At least it would make more sense to me if it was U]0,1[. Can anyone explain why the notation is so, and whether there are any other exceptions?

Answer 1

I don't know where you are getting this "Usually distribution notation is such that you give the name of the distribution, then its mean, and finally the variance" from.

As for the uniform distribution, the best way to imagine the uniform distribution is to know where it starts and where it ends. It gives you a quick idea of what the curve looks like.

The parameters of the normal distribution do too. The mean tells you were it is centered and the variance gives you an idea of the "spread".

So they are consistent in that sense. If I tell you the mean and variance of the uniform distribution, then you have to do some calculations to figure out what it looks like. Giving you the end points, you immediately know where it is.

However, if I think about $\text{Gamma}(n,\lambda)$, I don't think about the curve at all. I immediately understand it to be the sum of $n$ independent exponentials, each with rate $\lambda$. The expectation of such a random variable is $n/\lambda$ with varaince $n/\lambda^2$. Also, that's what it means to me, but this same notation means something else to other users.

There are more examples that you can look into on your own, like the Beta distribution.

So all in all, I don't believe they are consistent in how they are presented. But they consistent in that they do convey some useful information fast. But it should not be a surprise. Lots of notations and definitions are inconsistent across math and math textbooks.

Also, $U]0,1[$ is not common notation, specifically $]a,b[$. I know what you mean, but it is not universally used.

Answer 2

Are you from France or somewhere in Europe? In France, you use $]0,1[$ to denote the open unit interval. Some other countries use $(0,1)$. Regardless, $U$ can be regarded as a function taking two inputs and producing a mathematical object.

I find $U(0,1)$ much more intuitive than $U(0.5,1/12)$. I would say it is more convenient to be given the bounds of the interval than the mean and variance and having to figure out what the bounds are (though regardless of which direction we go, it is a trivial computation).