Discrete random uniform distribution

Standard series results

It’s important to remember our standard series results when working with discrete random uniform distributions. The main two are these ones:

Transformations

Another reminder of how we transform discrete random probabilities.

If is a transformation of the DRV :

Rectangular distribution

If we have something that looks like a uniform distribution, but it’s a rectangle instead of vertical lines of the same height, then it’s not a uniform distribution, but instead a rectangular distribution.

You should be able to tell: the graphs of rectangular distributions are, well, rectangles.

Notation for distributions

We can represent a probability distribution using a notation like this:

That means that is an A distribution (whatever A may be) and we need to know the parameters: , , etc. to be able to work with it.

Uniform distribution notation

This means that is a discrete random variable which follows a uniform distribution, from to .

Example: rolling a die

For a normal, fair, 6-sided die, our distribution is:

Let’s say we were to write a table of our probabilities for this distribution. It would look like this:

	1	2	3	4	5	6

Let’s calculate our expected value and variance for the distribution:

General formula

We’ve calculated the expected value and variance for a specific example, but we can do it more generally. Let’s say that for .

We can put this into a table to visualise it:

	1	2	3	…
				…		1
				…
				…

To calculate our expected value:

• So the expected value of a discrete random variable which follows a uniform distribution from to is .

Now, our variance. This one’s a little more complicated:

For a discrete random variable which follows a uniform distribution from to :