Let $X$ and $Y$ be random variables that are both exponentially distributed with different parameters. I dont get the idea how can they be dependent then... and we can only manipulate one parameter in exponential distribution so when that parameter makes random variables dependent and when independent?
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$\begingroup$ There is a simple test for independence. Have you tried calculating $E[XY]$, $E[X]$, and $E[Y]$ for two exponential random variables $X$ and $Y$? $\endgroup$– TitusCommented Jan 15, 2015 at 8:07
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$\begingroup$ But I dont know the distribution of XY when i dont know if they are independent $\endgroup$– luka5zCommented Jan 15, 2015 at 8:11
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$\begingroup$ I'm sorry - I misread what you mean to prove. From the context of the problem statement I would assume that the variables are independent to begin with. Can you give more background on where this problem comes from? $\endgroup$– TitusCommented Jan 15, 2015 at 8:27
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$\begingroup$ Its the general problem. How can two exponentialyy RVs be dependent or independent takin into account their parameters $\endgroup$– luka5zCommented Jan 15, 2015 at 8:29
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$\begingroup$ By definition. For example, they could both be determined independently (2 samples of 2 r.v.s) or they could be determined by common noise: if $u$ is a uniform random variable on $[0,1]$ we could have $X$ and $Y$ satisfy $F_X(X) = \int_0^{X} dF(x) = u$ and $F_Y(Y) = \int_0^{Y} dF(y) = u$. In both cases the marginals are the appropriate exponential random variables. $\endgroup$– TitusCommented Jan 15, 2015 at 9:18
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The distribution of $(X,Y)$ is a joint distribution. Your first line is simply saying what the marginal distributions are. The marginal distribution does not, in general, determine the joint distribution (except in the case of independent RVs). Thus, specifying the marginals does not preclude them being related.
In other words, the marginal distributions do not contain any information about their dependence.