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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?
Time to Evolve 2
Age 16 to 18
Challenge Level 
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 .
Possible support
Students struggling to get started could try the easier,
related, question time to evolve .
