Piecewise integration: find the 75th percentile of this continuous random variable.

Question

The problem from a quiz of mine is as follows:

In general, I know how to find the $p$th percentile of a random variable $X$, given its probability density function $f(x)$. You simply solve for the value $x_p$ that satisfies the following equation:

$\int^{x_p}_{-\infty}{f(x)dx} = \frac{p}{100}$

When given just a single function that isn't broken up piecewise like this, I know how to proceed. But here's my issue with this problem:

How do I split up the integral into multiple pieces, considering the upper limit of integration is a variable?

Answer 1

You'll want to integrate each piece, left to right, until you find out which part of the domain contains the $75$th percentile. Then you can do your integral equation in that part.

To be more precise, the first thing to calculate is the integral of $f$ from $0$ to $1.3$. If that's more than $0.75$, then you can write down your equation to figure out how much of it you need to integrate to get $0.75$. On the other hand, if it's less, then it equals some $p$. Then you need to find out how far to integrate $f$ from $1.3$ to obtain an area of $0.75-p$.

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$$\int_{1.3}^t\frac{2-x}{0.7}dx = \left[\frac{20x}{7}-\frac{5x^2}{7}\right]_{1.3}^t=\frac{20t-5t^2}{7}-\frac{351}{140}$$

Setting this last expression equal to $\frac1{10}$ and simplifying, I get a solution at $$t=2-\frac{\sqrt{35}}{10}\approx 1.40839$$

Answer 2

You have to split the integral up, too. For example, we know that $\int_{-\infty}^0{f(x)dx}=0,$ and we know $\int_{0}^{1.3}{f(x)dx}=.65,$ if I've done that right, so now you only need to find $x_p$ such that $\int_{1.3}^{x_p}{\frac{2-x}{.7}}=.10$