This problem gives an opportunity to explore the properties of pdfs using the mathematics of sectors of circles.

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.

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?

Try Into the Exponential Distribution.

Point students in the direction of the formula for the segment of a circle and that the area must be 1.

Normal distribution. Probability. Estimation of areas under curves. Definite and indefinite integration. Probability density functions. Averages. Mathematical reasoning & proof. Mathematical modelling. Probability distributions, expectation and variance. Functions and their inverses. Maths Supporting SET.