This problem makes use of
the distribution makerinteractivity
Why do this problem?
This problem requires students to invent a distribution to satisfy
certain criteria. In doing so, they will strongly reinforce their
understanding of the meaning of distributions as a whole. Students
will have to think creatively and really engage with the
relationship between a distribution and the probabilities in order
to fully solve the problem. The fact that universal order exists
underlying all random processes (i.e. the inequalities) is
interesting; instead of a collection of diverse results,
distributions become part of a coherent whole.
A main part of this challenge is understanding the meaning of
the question. Spend time with the group exploring what they feel
that question is asking and encouraging them to work through the
meaning of the parts of the question. Once they have a feel for the
question they might want to proceed by:
- Looking at the inequalities for some known distributions (such
as the binomial or the normal)
- Investigating other simple probability distributions (such as
that obtained on the roll of a die)
- Creating distributions 'randomly' using the distribution maker
and investigating the inequalities for the probability
The distribution maker interactivity will help students to see
probabilities for different distribution results.
- What is the problem asking?
- How can we use the condition that (??) is a whole number?
- Are then any obvious distributions to try out?
As a challenge, try to create a distribution which violates
the inequalities. How close can you get? Does this give you a feel
for why the inequalities hold? How close to the inequalities do
distributions which the class already know about (such as binomial
or normal) get?
You might like to refer keen students to the central limit theorem in probability,
which shows that on average distributions tend to the normal.
Suggest for the first part that students tabulate the probability
that X exceeds 1, 2, 3, 4, 5 and 6 for the roll of a single die.
Repeat for the roll of 2 dice (these situations are pre-programmed
into the distribution maker). Using these explicit results they can
explore the inequalities.