### Why do this problem?

This
task involves investigating pdfs and modelling though the
uniform distribution. It combines some theoretical analysis of mean
and variance and can be used to create an intuition that the sum of
random variables tends to a normal distribution with the aid of a
spreadsheet simulation. It is a common misconception that the sum
of uniform random variables is also uniform and this problem will
allow students to see that this is not the case in a meaningful
context.

### Possible approach

This problem could easily be done individually, but questions
concerning the soundness of the modelling assumptions would suit
themselves to group discussion.

The spreadsheet simulation really brings this task to life.
Although building such a spreadsheet is an interesting learning
task, you can download a ready-made Excel 2003 spreadsheet for this
purpose

here
.

One of the main obstacles to getting into this problem is the
technical formalism. Although the symbolism is very natural,
learners will need to be encouraged to read the question
carefully and think how the
words relate to the symbols and why this setup makes sense.

### Key questions

What is Var$(X+Y)$? What is Var$(aX)$?

Why is the sum of two uniform random variables not uniform?

Could we use a variant of this idea to model the time between the
births and deaths of humans?

### Possible extension

Students, for any problem, can always:

- Explore the ideas discovered on the internet or through
NRICH articles .

It is often surprising to people that there is no nice analytic
expression for the sum of a set of uniform distributions. The web
link mentioned in the foot of the problem would lead the interested
student to many nice results.

### Possible support

Students struggling to get started could try the easier,
related, question

time to evolve .