# Determining a PDF from its moments

I have a probability density function, $f_{X,Y}(x,y)$. I can calculate moments of $f$ easily, namely, I can calculate $\int\int x^ay^bf_{X,Y}(x,y)\,dx\,dy$ over the range of $X$ and $Y$, for any particular $a$ and $b$ I like.

If I calculate first and second moments, I could approximate $f$ with a gaussian. Suppose I calculated the third and/or fourth moments too. What might I approximate $f$ by, taking into account these extra moments?

[NB: if it helps, I know that $(X,Y)$ are bound within a triangle (say, $X\geq 0$, $Y\geq 0$, $X+Y\leq 1$, but it's up to me). I also expect that $f$ has the form of an algebraic function of $x$ and $y$, but I'm happy to ignore this]

• Calculate the skewness and excess kurtoses and then depending on the results choose likely distributions. For instance if skewness=0, or at least more or less zero, then this eliminates many possibilities. Sep 19, 2017 at 6:44
• You may find this useful Sep 19, 2017 at 7:37
• @caverac did you just link back to my own question? Sep 21, 2017 at 12:36
• :) Sorry, for some reason I copied the wrong link Sep 21, 2017 at 12:39
• This looks like a two-dimensional moment problem. Oct 22, 2018 at 0:11