Why do this problem?
is a sequence of activities based around the exponential
distributions. The aim is to draw the learner into an understanding
of the properties of pdfs without requiring too many complicated
calculations: it uses and will reinforce ideas about functions,
integration and areas. It will also suit self-motivated independent
The questions start off very simply and would lead to good
group discussion. You may wish to use only parts of the question
and it would be of value when students have encountered pdfs but
have not necessarily had much experience with their manipulation.
Some of the questions naturally ask 'why' a result is true.
Learners should be encouraged to think about these, but not get too
concerned if they can't come up with a $100\%$ watertight answer.
The key is to understand that the results intuitively must be true
and get into calculating and estimating probabilities from a
What do you know about the areas under a pdf?
Are there any turning points? What does this tell you?
Can students analyse the different areas formed by these three
Encourage learners to rely on their intuitive understanding of
integration in terms of area. Alternatively, focus on the last two
parts of the question as a discussion.