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?
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.
- Solved – Why is pattern recognition often defined as an ill-posed problem
- Solved – Suppose X and Y are independent Poisson random variables with respective parameters $lambda$ and 2$lambda$. Find $E[Y-2*X|X+Y=10]$
- Solved – How to estimate a percentile of a sum of two independent distributions
- Solved – the distribution of the Euclidean distance between two normally distributed random variables
- Solved – Derive the distribution of the ANOVA F-statistic under the alternative hypothesis