Why do this problem?
will require learners to engage with the key properties
of pdfs and cdfs and to understand how these relate to actual
functions. The reasoning required is quite sophisticated, although
the actual answer might be relatively short.
The problem would be of particular value either at the start
or the end of a body of work on pdfs and cdfs.
The first obstacle to overcome is to understand properly the
problem, as it might be more formally stated than students are used
to. Once this is done, students should be encouraged to think about
the properties of pdfs and cdfs, and then to start addressing the
problem. As there is little 'calculation' required, encouragement
and discussion will most likely be of use.
Part of the problem is showing or explaining why it is that certain forms which
would work as a cdf cannot be used as pdf, or
vice-versa, so the emphasis should be on clear reasoning.
It is important to be aware the the problem does not require
any assessment of the type of probability process underlying the
proposed pdfs and cdfs: it can, and should, be done entirely
What is the single most important constraint that a function
being used as a pdf must satisfy?
What are the constraints on a function which is to be used for
Can you describe a probability process which would give rise
to the pdfs and cdfs you come up with in the problem?
Look at some concrete examples of pdfs and cdfs. In each
specific case, what is it that prevents the pdf being used as the
cdf, or vice versa?