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Circle pdf
Age 16 to 18
Challenge Level 
Why do this problem?
This problem gives an opportunity to explore the properties of pdfs using the mathematics of sectors of circles.
Possible approach
There are two main parts to this problem.
The first is to understand why certain shapes are firstly valid pdfs and secondly how they satisfy the technical requirement of the question. This would benefit from a discussion approach.
The second part, calculating the maximum value, will lead students into the mathematics of sectors and segments of circles.
Key questions
What properties must a pdf have?
How would the requirements of the questions relate to a graph?
In order to obtain the maximum possible value for the case of the circle of radius $1$, what do we know about the arc?
Possible extension
Try Into the Exponential Distribution.
Possible support
Point students in the direction of the formula for the segment of a circle and that the area must be 1.
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