Expected value with two random variables

Question

A line segment AB of length 1m is broken in two at a random point P where the length of AP has the following probability density function:

$f(x)=6x(1-x), 0<x<1$

A point Q is uniformly selected from AP at random. If the length of AQ is found to be 0.5m, find the expected length of AP.

Please help solve this question. Thanks a lot :D

Answer 1

This is just a conditional probability question, right? Since you know that the coordinate $P\geq 0.5$, you have to rescale your probability distribution $f(x)$ on the new domain $0.5<X<1$. So your new probability distribution given that $x>0.5$, $f_{X>0.5}(x)$, should be:

$$f_{X>0.5}(x)=\frac{f(x)}{\int_{0.5}^1 f(x)dx}$$

To compute your expected value of $X$ now, given $X>0.5$, it suffices to calculate:

$$E[X | X>0.5]=\int_{0.5}^{1}x\cdot f_{X>0.5}(x)dx.$$