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Age 16 to 18 Challenge Level:

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.