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Given a probability density function find the mean, median and mode of the distribution.

Scale Invariance

By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

Into the Exponential Distribution

Get into the exponential distribution through an exploration of its pdf.

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.