We observe the input samples as $Z = X+Y$, where distribution of $Z$ could be estimated by histogram of samples and $X,Y$ are two independent random variables. One of the variable was known following exponential distribution (as $X sim Exp(lambda)$).

I want to get the parameter of $X$, $lambda$, and probablity density of $Y$ with its parameters also. How to decompose the random variable $X+Y$?

Further more, what if the distribution of $X$ is unknown?

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#### Best Answer

The problem is ill-posed. There is no unique solution if you dont specify Z. Think about it. Pick any Y with any distribution you choose. X+Y will have some distribution that depends on what you chose for Y. So if you specify Z then Y is determined but two distinct choices for Y will give two different distributions for Z. There are infinitely many solutions to your problem.

Now given that of course your second problem is also ill-posed. Just knowing that X and Y a re independent tells you almost nothing about Z. Take X and Y normal and Z will be normal. If X and Y are identical distributed Cauchys Z will be Cauchy. Two independent chi squares lead to Z be chi-square. So there are three solutions to problem 2 and that is only scratching the surface

In problem 1 you need to know Z's distribution to determine Y's. In problem 2 you cant get Z without specifying the distributions of X and Y.

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