Given a probability density function find the mean, median and mode of the distribution.
Problem
Getting Started
The area under the graph of the probability density function between $x=a$ and $x=b$ gives the probability that the outcome is between $a$ and $b$ so the total area under the graph must be $1$, in this example for $x$ between $0$ and $3$.
To find the median we have to find the value $t$ such that the area under the graph for $0\leq x \leq t$ is $0.5$. You will have to find the roots of a cubic equation (which you should be able to factorise) in this example and then identify the root which lies in the required interval.
Student Solutions
Here is another excellent solution from Andrei at Tudor Vianu National College, Romania:
Teachers' Resources
The advantage of using a continuous probability density function to model a set of statistical data is that instead of having to do calculations with all the separate values we can use calculus to find the required results.
The median is the point $t$ in the interval such that $$\text{Prob}\{0 \leq X \leq t\} = \text{Prob}\{t \leq X \leq 3\}$$
The mean of the distribution is $m$, where $$m=\int_0^3x\,\rho(x)\,dx.$$
In a discrete distribution the mode is the value that occurs most frequently. In a continuous distribution it is the value where the probability density function takes its maximum value.