# Given the pdf of Z = X+Y, what's the pdf of X if X&Y are IID

A common problem in statistics is, "Given that X is distributed according to the pdf f(x) and Y is distributed according to the pdf g(y), what is the pdf h(z) of Z = X + Y?" The answer is the convolution of f(x) and g(y):

$$h(z)=(f*g)(z)=\int_{-\infty}^\infty f(z-t)g(t) dt = \int_{-\infty}^\infty f(t)g(z-t) dt$$

In my research I have stumbled across the reverse problem given that X and Y are IID (identically and independently distributed): "Given that the random variable Z = X+Y has a known pdf h(z), what is the pdf f(x) if it is known that X and Y are IID?"

Does anyone have insight to how this problem could be solved? Is there such a thing as a deconvolution operator that could help me? Thanks in advance.

## Answers and Replies

mfb
Mentor
If you know Z with arbitrary precision, a Fourier transform should help.
Or calculate all the moments of the distribution.

• ecneicScience
If you know Z with arbitrary precision, a Fourier transform should help.
Or calculate all the moments of the distribution.

Exactly!!! I never actually thought I'd use those convolution theorems I learned in my FT class!

We know
$$h = f*g$$
Taking the Fourier transform
$$\mathcal{F}\{f*g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\}$$
We know f = g, so
$$\mathcal{F}\{f*g\} = \mathcal{F}\{f\}^2$$
Rearranging,
$$f = \mathcal{F}^{-1} \bigg\{ \sqrt{ \mathcal{F} \{ f*g \} } \bigg\}$$
Putting in h,
$$f = \mathcal{F}^{-1} \bigg\{ \sqrt{ \mathcal{F} \{ h \} } \bigg\}$$

Thank you for that insight. Hopefully my data is not too sparse so that I can perform a decent FFT.

Could you please elaborate how calculating moments could help?

mfb
Mentor
If you know all moments, you can reconstruct the function, and you can find relations between the moments of Z and X.

A discretized version can always have problems with high frequencies, but I guess there are ways to suppress them.

• ecneicScience
If you know all moments, you can reconstruct the function

That is not necessarily true. But I guess you can assume that the distribution of the OP is nice enough for it to be true.

• ecneicScience
mfb
Mentor
For all functions where you have some chance to reconstruct it experimentally ;).

• ecneicScience
If you know all moments, you can reconstruct the function, and you can find relations between the moments of Z and X.

A discretized version can always have problems with high frequencies, but I guess there are ways to suppress them.

Is there a theorem you could refer me to that describes this process of constructing a pdf from its moments?

mfb
Mentor

That article didn't seem that helpful or it went way over my head. What I had in mind when you said a pdf could be reconstructed from its moments, is that the pdf could be approximated as some polynomial, where each term is somehow a function of the first N moments, and that the reconstruction of became increasingly accurate with increasing N. I couldn't find anything that mentioned this in that article.

mfb
Mentor
The references given there might help.

Checking Wikipedia, the references given there, and google results with similar keywords is really not magic, and something that could be expected before asking questions.

Checking Wikipedia, the references given there, and google results with similar keywords is really not magic, and something that could be expected before asking questions.

Thanks for the tip.

Can you just look at this as computing the probability of X and Y, eg their union?