A cumulative probability, \(P(X \le x)\), can be found by integration. It is sometimes useful to work in the opposite direction — given a cumulative probability, what is the corresponding value of \(x\)?
Definition
The \(p\)'th quantile of a continuous distribution is the value, \(x\), such that
\[ P(X \le x) \;\; = \; \; p \]When \(p\) is expressed as a percentage, the value is called the \(100p\)'th percentile.
Definition
These three values split the probability density function into four equal areas.
Question
What are the median and quartiles of the \(\RectDistn(1, 5)\) distribution?
(Solved in full version)
The next example is a little harder.
Question
Find a formula for the \(p\)'th quantile of the exponential distribution with probability density function
\[ f(x) = \begin{cases} 4\;e^{-4x} & \text{for } x \ge 0\\[0.2em] 0 & \text{otherwise} \end{cases} \](Solved in full version)