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

1. Apr 15, 2016

### ecneicScience

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.

2. Apr 15, 2016

### Staff: Mentor

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

3. Apr 15, 2016

### ecneicScience

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?

4. Apr 15, 2016

### Staff: 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.

5. Apr 15, 2016

### micromass

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.

6. Apr 15, 2016

### Staff: Mentor

For all functions where you have some chance to reconstruct it experimentally ;).

7. Apr 15, 2016

### ecneicScience

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

8. Apr 16, 2016

### Staff: Mentor

9. Apr 16, 2016

### ecneicScience

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.

10. Apr 16, 2016

### Staff: 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.

11. Apr 16, 2016

### ecneicScience

Thanks for the tip.

12. Apr 21, 2016

### OrangeDog

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