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Math Help - Continous random variables questions

  1. #1
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    Continous random variables questions

    I have having difficulties with this question

    I have attached my question.

    any help is appreichated

    thanks
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  2. #2
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    Ever heard of the law of the total probability?
    \mathbb{P}(X+Y\leq a)=\mathbb{E}[\mathbb{P}(X\leq a- Y|Y)]
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  3. #3
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    could you please explain that law by applying to my question.

    thanks
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  4. #4
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     \mathbb{P}(X\leq 1/2 -Y |Y)=[1/2-Y]^+ where [x]^+ is the maximum between x and 0 (because if Y is greater than 1/2, we cannot have a negative probability, and the prob of X being less than a negative number is 0).

    Now
    <br />
\mathbb{E}[1/2-Y]^+=\int_0^1 \max\{1/2-y,0\}\mathbb{P}(Y\in dy)=\int_0^{1/2} (1/2-y) dy<br />
    you can complete the rest.
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