# Characterization of joint probability density function of independent random variables

Let $$f(x,y)$$ be a joint probability density function (pdf) of two random variables $$X$$ and $$Y$$. To check whether $$X$$ and $$Y$$ are independent, we can compute the marginal densities and check if their product equals $$f(x,y)$$.

My question is: Is there a characterization of functions that are pdf of independent random variables, i.e., can we "easily" decide whether $$X$$ and $$Y$$ are independent without determining the marginal density functions?

If this is not the case, is a characterization known when we restrict $$f(x,y)$$ so simple functions, say, polynomials?

• It is enough if you know that $f(x,y)=g(x)h(y)$ for some non-negative measurable functions $g$ and $h$. Nov 19, 2019 at 12:01

As Kabo Murphy points out, the test is whether $$f$$ factors as $$f(x,y)=g(x)h(y)$$.
A test for that is that, for almost all $$x_1,x_2,y_1,y_2$$, the identity $$f(x_1,y_1)f(x_2,y_2)=f(x_1,y_2)f(x_2,y_1)\tag{*}$$ should hold.
This might be useful if you have a very complicated analytic expression for $$f(x,y)$$, for which the factorization is not visible but (*) can be checked algebraically or numerically.
A caveat: density functions are only defined "almost everywhere": modify a density function for a measure-$$0$$ set of argument values and it counts as the same density function. So (*) must hold for almost all values of the $$x_i$$ and $$y_j$$.
Added later. Another test is that $$\frac \partial{\partial x} \frac \partial{\partial y} \log f(x,y)$$ must vanish for all $$x,y$$.