I have a question about joint distributions but couldn't find the solution:

Suppose that $X$ and $Y$ are two random variables and their joint pdf is given by $$f_{XY}(x,y)=cxy(1-x-y), \qquad0<x<1, \,0<y<1,\,0<x+y<1$$ for some $c>0$. Find the covariance of $X$ and $Y$.

Actually I couldn't do anything about the problem. Tried to find the marginal pdf's of $X$ and $Y$. But I couldn't understand the $0 < x+y < 1$ part. How can use this information in this problem.

I appreciate any help. Thank you in advance.


Hint: The marginal pdf of $X$ is given by $$f_X(x)=\int_Y f_{XY}(x,y)dy=\int_{0}^{1-x}f_{XY}(x,y)dy$$ That is because you have the restriction $0<x+y<1$ i.e. $$-x<y<1-x \implies 0<y<1-x$$ since $-x<0$ and it is already stipulated that $y>0$.

  • $\begingroup$ Thank you.But i need to ask something.My way doesn't seem to be the right way.Now I know the marginal pdf's but I can not find the covariance with this information. $\endgroup$ – Zapdos Oct 19 '14 at 11:24
  • $\begingroup$ $Cov(X,Y)=E[XY]-E[X]E[Y]$. To find these expected values might be computanionally tedious but it should be straightforward from the definitions in your notes $\endgroup$ – Jimmy R. Oct 19 '14 at 11:25
  • $\begingroup$ I think I get it.So I am going correct,I need these marginals anyway.But computation will be really long. $\endgroup$ – Zapdos Oct 19 '14 at 11:42

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