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Age 16 to 18 Challenge Level:

Why do this problem?

This problem gives an opportunity to practise numerical integration in the context of probability distributions. It will really allow students to get into the meaning of probability density functions in terms of areas and probabilities. Instead of simply requiring an explicit calculation, students will need to engage with decisions concerning limits and integration.

Possible approach

The first stage of the problem is to realize that a numerical integration is needed to calculate the mean. Once the class has realised that this is the case, they will need to start to perform the integrations. This will require various choices as there are many ways in which this can be done. To facilitate this, you might like to print off copies of the graph for students to draw on.

Key questions

How do we relate a probability density function to a probability?
How do the two graphs relate to each other?
What is the graphical interpretation of an integral?
How important will the effect of the second graph be?
What happens for values larger than $20$? Are these values relevant?

Possible extension

How might you try to estimate the variance for these distributions numerically?

Possible support

First try to show that numerically the area under the red curve is 1. You can then use the decomposition into rectangles and trapezia to try to work out the mean.