This question arises from the context of computing the distribution of total execution time with an underlying graph of tree. For instance, we can model the execution times of all the nodes on the graph by independent random variables whose distributions come from a certain parametrized family, such as the normal distributions. Then the total execution time can be modeled as the maximum and sum of these random variables. The cdf of the maximum of two independent distributions is the product of their cdfs, whereas the pdf of the sum of two independent distributions is the convolution of their respective pdfs.
Update: explanation of notation. $F:\mathbb{R}_+ \to [0,1]$ is a smooth cumulative distribution function. Thus it is non-decreasing. Examples include $F(x) = 1- e^{-\lambda x}$, the cdf of an exponential random variable of parameter $\lambda$.
By $F^2$ I simply mean the square of $F$; $F'$ stands for the derivative of $F$ with respective to its only variable. $(F \ast F)(x):= \int_{-\infty}^\infty F(t) F(x-t) dt$ is the convolution of $F$ with itself. Furthermore $F^{\ast k} = F \ast F \ast \ldots \ast F$ for $k$ times. I am implicitly assuming $k$ is an integer, but if $F$ comes from a continuously parametrized family, such as the Gaussians, then even non-rational $k$ makes sense.