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Math Help - probability distribution and density function

  1. #1
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    probability distribution and density function

    Hi,

    How do you determine the probability distribution function F of the stochastic variable U:= X+Y where X and Y are independent uniformly distributed over [0,1]?

    My thought was to solve this by first determining the density function f of U
    So f wil be the convolution product of the density functions of X and Y
    So f(u)=(g * h)(u) where g and h are the density functions of X and Y

    Then Prob(U<= c) = F(c) = integral (from 0 to c) f(x)dx

    Is this the correct way of determining F?

    And if I want to find the density function of Z:=X-Y is that also possible by using the convolution of the density function of X and -Y? Is -Y uniformly distributed?

    I hope anyone can help me out!

    Thanks!

    Kind regards,
    Joolz
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  2. #2
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    Quote Originally Posted by Joolz View Post
    Hi,

    How do you determine the probability distribution function F of the stochastic variable U:= X+Y where X and Y are independent uniformly distributed over [0,1]?

    My thought was to solve this by first determining the density function f of U
    So f wil be the convolution product of the density functions of X and Y
    So f(u)=(g * h)(u) where g and h are the density functions of X and Y

    Then Prob(U<= c) = F(c) = integral (from 0 to c) f(x)dx

    Is this the correct way of determining F?

    And if I want to find the density function of Z:=X-Y is that also possible by using the convolution of the density function of X and -Y? Is -Y uniformly distributed?

    I hope anyone can help me out!

    Thanks!

    Kind regards,
    Joolz
    Yes convolution is what you use to find the density of the sum of two RVs,

    If Y ~U(0,1), then -Y ~U(-1,0)

    CB
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  3. #3
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    Quote Originally Posted by Joolz View Post
    Hi,

    How do you determine the probability distribution function F of the stochastic variable U:= X+Y where X and Y are independent uniformly distributed over [0,1]?

    My thought was to solve this by first determining the density function f of U
    So f wil be the convolution product of the density functions of X and Y
    So f(u)=(g * h)(u) where g and h are the density functions of X and Y

    Then Prob(U<= c) = F(c) = integral (from 0 to c) f(x)dx

    Is this the correct way of determining F?

    And if I want to find the density function of Z:=X-Y is that also possible by using the convolution of the density function of X and -Y? Is -Y uniformly distributed?

    I hope anyone can help me out!

    Thanks!

    Kind regards,
    Joolz
    http://www.dartmouth.edu/~chance/tea...k/Chapter7.pdf Read page 7.
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