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    Expected Value calculation




    I have 2 random variables U and V uniformly distributed on [0, 1]. Their joint cumulative distribution function F(u, v) = min (u,v) on [0, 1] x [0, 1].

    Therefore I cannot calculate their probability density function by classic differentiation of F(u, v). How can I calculate such quantities as the Expected Value of U+V or the Expected Value of U*V?

    Thank you in advance for your help

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    Re: Expected Value calculation

    Quote Originally Posted by 1stone View Post
    I have 2 random variables U and V uniformly distributed on [0, 1]. Their joint cumulative distribution function F(u, v) = min (u,v) on [0, 1] x [0, 1].

    Thank you in advance for your help

    Let me just ask - for clarification - is your scenario different from having U and V as independent R.V.s that are uniformly distributed on [0 , 1], where Z=min(U,V) and you want to know what the pdf associated with Z is?

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    Re: Expected Value calculation

    Yes, my scenario is different, as in my case U and V are not independent uniform r.v.

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    Re: Expected Value calculation

    http://en.wikipedia.org/wiki/Copula_...ula_boundaries

    I guess the joint density does not exist, because U = V ~~ \mathrm{a.s.},
    they are perfectly correlated, and thus the random vector is not 2-dimensional.

    \Pr\{U \leq u, V \leq v\} = \min\{u, v\} = \Pr\{U \leq \min\{u, v\}\}

    = \Pr\{U \leq u, U \leq v\} = \Pr\{V \leq u, V \leq v\} ~~ \forall (u, v) \in [0, 1]^2

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    Re: Expected Value calculation

    Thank you, BGM! I conclude that the expectations I am looking for cannot be calculated by usual integration of the joint pdf. Still, is there any way to calculate them?

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    Re: Expected Value calculation

    If you know U = V ~ \mathrm{a.s.},

    then it should not be hard to see

    E[U + V] = 2E[ U], E[UV] = E[U^2] etc.

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    Re: Expected Value calculation

    Also note that we didn't need to know anything about the joint distributions to figure anything out E[U + V]. We do need the joint distribution to say something about E[UV] though.

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    Re: Expected Value calculation

    Thank you! Now I can calculate these expected values.

    As U = V a.s., would it be correct to calculate the expected value of any continuous function f(U, V) by integrating f(u, u) from u=0 to u=1?

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    Re: Expected Value calculation

    Quote Originally Posted by 1stone View Post
    Thank you! Now I can calculate these expected values.

    As U = V a.s., would it be correct to calculate the expected value of any continuous function f(U, V) by integrating f(u, u) from u=0 to u=1?
    It should be alright.

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    Re: Expected Value calculation


    in matlab code:

    for i=1:length(observed(:,1))
    for j=1:length(observed(1,: ))
    expected(i,j)=sum(observed(:,j))*sum(observed(i,: ))/sum(sum(observed));
    end
    end

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