Why do this problem?
This problem offers a fascinating exploration into probability
density functions for real world data. Whilst the individual steps
are quite simple, the problem draws together many strands from
distribution theory. The results can be tested on any set of data
from any geography book, giving an interesting relevance to the
The first obstacle to overcome is that of notation: can the
students understand what is being asked?
The question involves little computation but requires clear
thinking of the ideas. This might be facilitated in a group
discussion, but might also require individual work.
- If a function is to be a probability density function, what is
the major property it must possess?
- What ranges of values will start with a digit $1$?
Consider carefully why this problem involves 'scale invariance'.
Consider the restriction of scale invariance on real world data.
Which sets of real world data do you think will be modelled by this
Skip the first part and provide students with the scale invariant
functions. Also, first use the range 1< x < 10 in the last
part of the question.