Conditional distribution and density function of a Uniform distribution

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let's say I have U~Uniform(0,1) and W~Uniform(0,U), I have a few questions about it



  • How can I determine the conditional distribution of W given U? i.e.
    FW|U(w|u) = P[W ≤ w|U = U]

  • How can I find the density function of W given U

  • Finding the unconditional density function of W

For the first part, by bayes:



$$P(U > s | U > t) = fracPleft[U > scap U > t right]P(U > t)$$



so it follows



$$Pleft[U > scap U > t right]=P(U > s)$$



But I am not sure how to continue or finish if this is correct



For the second question, I assume I have to use the formula for a conditional distribution, however I think this is wrong as well



∫∫fXY(x,y)dxdy=1



For the last questions, I am unsure about how to answer it at all.



Any help is appreciate, thank you!




proof-explanation uniform-distribution




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    up vote
    0
    down vote

    favorite
    1












    let's say I have U~Uniform(0,1) and W~Uniform(0,U), I have a few questions about it



    • How can I determine the conditional distribution of W given U? i.e.
      FW|U(w|u) = P[W ≤ w|U = U]

    • How can I find the density function of W given U

    • Finding the unconditional density function of W

    For the first part, by bayes:



    $$P(U > s | U > t) = fracPleft[U > scap U > t right]P(U > t)$$



    so it follows



    $$Pleft[U > scap U > t right]=P(U > s)$$



    But I am not sure how to continue or finish if this is correct



    For the second question, I assume I have to use the formula for a conditional distribution, however I think this is wrong as well



    ∫∫fXY(x,y)dxdy=1



    For the last questions, I am unsure about how to answer it at all.



    Any help is appreciate, thank you!




    proof-explanation uniform-distribution




    share|cite|improve this question





















      up vote
      0
      down vote

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      1









      up vote
      0
      down vote

      favorite
      1






      1





      let's say I have U~Uniform(0,1) and W~Uniform(0,U), I have a few questions about it



      • How can I determine the conditional distribution of W given U? i.e.
        FW|U(w|u) = P[W ≤ w|U = U]

      • How can I find the density function of W given U

      • Finding the unconditional density function of W

      For the first part, by bayes:



      $$P(U > s | U > t) = fracPleft[U > scap U > t right]P(U > t)$$



      so it follows



      $$Pleft[U > scap U > t right]=P(U > s)$$



      But I am not sure how to continue or finish if this is correct



      For the second question, I assume I have to use the formula for a conditional distribution, however I think this is wrong as well



      ∫∫fXY(x,y)dxdy=1



      For the last questions, I am unsure about how to answer it at all.



      Any help is appreciate, thank you!




      proof-explanation uniform-distribution




      share|cite|improve this question











      let's say I have U~Uniform(0,1) and W~Uniform(0,U), I have a few questions about it



      • How can I determine the conditional distribution of W given U? i.e.
        FW|U(w|u) = P[W ≤ w|U = U]

      • How can I find the density function of W given U

      • Finding the unconditional density function of W

      For the first part, by bayes:



      $$P(U > s | U > t) = fracPleft[U > scap U > t right]P(U > t)$$



      so it follows



      $$Pleft[U > scap U > t right]=P(U > s)$$



      But I am not sure how to continue or finish if this is correct



      For the second question, I assume I have to use the formula for a conditional distribution, however I think this is wrong as well



      ∫∫fXY(x,y)dxdy=1



      For the last questions, I am unsure about how to answer it at all.



      Any help is appreciate, thank you!





      proof-explanation uniform-distribution





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Aug 3 at 17:04









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