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Math Help - Problem Involving Calculating the probability given the marginal density functions!

  1. #1
    Junior Member
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    Problem Involving Calculating the probability given the marginal density functions!

    Hello,

    I'm trying to solve this problem:

    If X and Y are independent and f(x) = 1/2 when 0<X<2 and f(y) = 2Y when 0<Y<1, find P(X<2Y).

    My attempt:

    As X and Y are independent, f(x,y) = Y when 0<X<2 and 0<Y<1.

    I integrated Y first with respect to X from 0 to 2Y, then I integrated this with respect to Y from 0 to 1, which gives 2/3 for the answer.

    Am I right?
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  2. #2
    MHF Contributor
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    Re: Problem Involving Calculating the probability given the marginal density function

    If SC313.

    That approach is spot on, but you need to consider X/2 < Y in which the order of integration is 0 to Y/2 for X and 0 to 1 for Y.

    Changing the order of integration gives 0 to 1/2 for X and 0 to 1 - 2X for Y.

    Integrating this with respect to dy with limits (0 to 1-2x) first gives (1/2)*(1-2X)^2 and integrating this over dx with limits (0 to 1/2) gives 1/6.

    Double check my calculations for the integration limits, but the approach you had to setup the PDF is right (and if I'm wrong about the integration please inform).
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