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
Yes, my scenario is different, as in my case U and V are not independent uniform r.v.
http://en.wikipedia.org/wiki/Copula_...ula_boundaries
I guess the joint density does not exist, because ,
they are perfectly correlated, and thus the random vector is not 2-dimensional.
Dason (06-17-2011)
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?
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.
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?
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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