PDF stories

Invent scenarios which would give rise to these probability density functions.

Age

16 to 18

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

Consider the following curves, with scales chosen so as to make their integrals between the endpoints equal to 1. Curves with a finite domain are indicated with a vertical line, and the axes are assumed to intersect at the origin. Make up some statistical situations, either from mathematics or from science, which would plausibly result in one of these curves as a probability density function.

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Keep your eyes open for curves in other areas of mathematics.Would they be potential candidates (ignoring scale) for probability density functions? What mathematical descriptions might accompany them? What scientific situations might they describe?

Pure maths extension: Find algebraic expressions for curves such as these, with parameters and domain chosen so as to make their integrals equal to 1 (exactly or approximately, as approriate)

Teachers' Resources

Why do this problem?

This problem is a good vehicle for fostering thinking about probability density functions and getting some talk into the statistics classroom. It could be used alongside a traditional introduction to the concepts to probability density functions or as a discussion-based consolidation. It can be approached from various levels of sophistication: all students should be able meaningfully to engage with it, and you can come back to the ideas at several points during a course of statistics.

Possible approach

To use this task you will need to be comfortable with the idea that mathematical discussions need not necessarily be based around the use of algebra and that a 'complete' answer might well be beyond both teacher and class. Much of school statistics can be quite mechanical so encourage and reassure students that clear statistical reasoning need not follow a strict algorithm or procedure.

One approach (harder) is to ask students to work secretly in pairs to attempt to create a 'pdf story' which would match a chart. After 10 minutes or so, ask for volunteers to describe the story that they are most proud of and see if the rest of the class can work out the intended match. If there is disagreement, come to some class decision as to whether there is indeed a match to the story. If (as is hoped) the class guesses correctly, spend a few minutes discussing how well the story matches the curve of the pdf. This is an important aspect of statistical modelling at university and beyond, and really brings the subject of statistics to life. You can repeat with stories matching different curves either in the same lesson or over a period of a few weeks.

Another approach (easier) is to select a curve and attempt as a class to jointly craft a satisfactory pdf story (you will need to be happy to facilitate and guide the discussion as necessary). Once you have had a go at this, you might revert to the other approach described above.

Throughout, the pedagogic focus should be on clarity of explanation and conceptual understanding; this will take some time to develop so encourage students to persist through the 'um' and 'er' phases of explanation.

As students engage with the curves you might wish to use the pure maths extension and then attempt computations based on the algebraic expressions produced by the students.

Key questions

What areas of the charts correspond to events that are more likely to occur than others?

What is the probabilistic interpretation of key areas of the charts?

Does your explanation convince you? Why or why not?

Possible extension

Students could follow the ideas from the links contained in the problem.

(Very difficult) Invent a plausible story and algebraic expression for every curve.

Possible support

Focus on charts A, C, I and J only.

Or search by topic

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners