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

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

    Quote Originally Posted by math951 View Post
    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]
    $P(|X-\frac 1 2|>\frac 1 4) = 1 - P(|X-\frac 1 2|\le \frac 1 4)$
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    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.
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