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$(X_n)$ is a sequence of i.i.d real valued random variables, where the distribution of $X_n$ is assumed to be the exponential distribution with mean 1.
We then define $Z_n=X_{2n}X_{2n+1}$.

Now I have to show, that $(Z_n)$ is a sequence of i.i.d random variables. I think it is easy to see, that they are independent, but what are the argument that they are identically distributed?

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  • $\begingroup$ If you can show that the distribution of $Z_n$ does not depend on $n$, the you are done. $\endgroup$
    – Gordon
    Commented Oct 12, 2016 at 13:36

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identically distributed means that the distribution of $Z_n$ does not depend on $n$. In this, the distribution of $Z_n$ can be directly derived from the distribution of $X_{2n}$ and $X_{2n+1}$.

The distribution of $X_{2n}$ and $X_{2n+1}$ do not depend on $n$ since $X$ is iid, therefore the result follows.

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