# Thread: Let X be a continuous random variable with density function.

1. ## Let X be a continuous random variable with density function.

https://imgur.com/a/w6FEULR

P[ X - 1/2 | > 1/4]

how do we get from there to here: P[ X - 1/2 | <= 1/4]

2. ## Re: Let X be a continuous random variable with density function.

Originally Posted by math951
https://imgur.com/a/w6FEULR

P[ X - 1/2 | > 1/4]

how do we get from there to here: P[ X - 1/2 | <= 1/4]
$P(|X-\frac 1 2|>\frac 1 4) = 1 - P(|X-\frac 1 2|\le \frac 1 4)$

3. ## Re: Let X be a continuous random variable with density function.

$\left |X - \dfrac 1 2 \right| > \dfrac 1 4$

$X - \dfrac 1 2 > \dfrac 1 4 \bigcup X-\dfrac 1 2 < -\dfrac 1 4$

$X \in \left(-\infty,~\dfrac 1 4\right) \cup \left(\dfrac 3 4,~\infty\right)$

$\displaystyle \int_{-\infty}^{\frac 1 4}~f(x)~dx + \int_{\frac 3 4}^\infty~f(x)~dx =$

$\displaystyle \int_0^{\frac 1 4}~6x(1-x)~dx + \int_{\frac 3 4}^1~6x(1-x)~dx$

I leave you to compute those last two integrals.