# Circle pdf

What happens if this pdf is the arc of a circle?

A random variable $X$ has a zero probability of taking negative values but has a non-zero probability of taking values in the range $[0, a]$ for every $a>0$. The curve describing the probability density function forms an arc of a circle. Which of these are possible shapes (ignoring the scale) for the probability density function $f(x)$? Identify clearly the mathematical reasons, using the
correct terminology, for your answers.

Image

If the radius of the circle forming the arc of the pdf is $1$, what is the maximum value that the random variable could possibly take?

Which of the other arcs are possible candidates for probability density functions? Can you invent mathematical scenarios which would lead to these pdfs?

Think what properties probability density functions must
have.

You will need to use the formula for the area of a sector of a circle.

You will need to use the formula for the area of a sector of a circle.

### 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.