Let us define a random variable (RV) where and are two independent RVs. If and are probability density functions of and , then what can we say about , the pdf of ? A rigorous double “E” graduate course in stochastic processes is usually sufficient to answer this question. It turns out is the convolution of the densities and . See this (p.291) for more.

It is tempting to ask the counter question: if , then does it imply where and are independent RVs? I found the answer in this amazing text:

*Counterexamples in Probability* by Jordan Stoyanov. The book is an adorable compilation of some 300 counterexamples to the probability questions which might be bothering you during a good night-sleep.

So, the answer to my question is no. The counterexample is using the Cauchy distributions. It turns out that the convolution of two Cauchy distributions is always a Cauchy distribution whether and are independent or not.

Coming to think of the convolution of pdfs, our favorite website has a list of convolution of common pdfs.