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I was wondering about the following:

Let's say we have two random variables $X,Y$ that obey both Poisson's distribution. Now, if we take $X=Y$ then they are clearly dependent. But what happens if we say that $X$ and $Y$ are Poisson distributions with different parameters $\lambda_x \neq \lambda_y$?

Does this mean, that they are independent?

If this is not true: Is it true for any distribution, that if you have random variables with different parameters, then they are automotically independent?

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Hint: Let $X$ be the number of Poisson events in the first hour, and $Y$ the number of Poisson events in the first two hours. If $X$ has parameter $\lambda$, then $Y$ has parameter $2\lambda$. The two random variables are not independent.

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  • $\begingroup$ thank you. so presumably, my more general question is also not true? $\endgroup$ – user66906 Mar 6 '14 at 16:39
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    $\begingroup$ I have not thought about formulating and proving a precise version. One can produce artificial examples where is not true, for example by picking $X$ to be $a$ with probability $1$. I doubt there are natural examples. $\endgroup$ – André Nicolas Mar 6 '14 at 16:43
  • $\begingroup$ okay, many thanks. $\endgroup$ – user66906 Mar 6 '14 at 16:43
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Suppose $W$ and $X$ are independent Poisson-distributed random variables each with expected value $1$.

Let $Y=W+X$. Then $Y$ is Poisson-distributed with expected value $2$.

So $X$ and $Y$ are Poisson-distributed random variabes with different expected values, but they are certainly not independent.

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  • $\begingroup$ thank you. so presumably, my more general question is also not true? $\endgroup$ – user66906 Mar 6 '14 at 16:38
  • $\begingroup$ This counterexample is also a counterexample to the more general statement. $\endgroup$ – Michael Hardy Mar 6 '14 at 19:43
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No, whether or not the parameters are equal has nothing to do with whether or not they are independent. They can have the same parameter value and be dependent or independent and likewise if their parameter values differ.

The only ways to figure out whether or not they are independent is to check whether or not their joint distribution function (cdf) factors as a product of the marginal cdfs, or ditto for the probability mass or characteristic functions.

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