1. ## 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?

2. ## Re: Expected Value calculation

Originally Posted by 1stone
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].

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?

3. ## Re: Expected Value calculation

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

4. ## Re: Expected Value calculation

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.

5. ## The Following User Says Thank You to BGM For This Useful Post:

Dason (06-17-2011)

6. ## 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?

7. ## Re: Expected Value calculation

If you know ,

then it should not be hard to see

etc.

8. ## 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.

9. ## 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?

10. ## Re: Expected Value calculation

Originally Posted by 1stone
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.

11. ## 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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