Unimodality of the probability distribution of the extensive functional of samples of a random sequence

We establish a criterion for the unimodality of the probability distribution of a functional that is represented by the sum of a set of independent identically distributed random nonnegative variables ${\tilde x}_k$ with a random number of terms distributed according to Poisson. The general distribution of terms ${\tilde x}_k$ is concentrated on the interval $[0, 1]$ and is such that $\mathrm{Pr}\,\{{\tilde x}_k = 0\} \ne 0.$ Its absolutely continuous part is asymptotically close to a uniform distribution. We introduce the concept of smoothing functions and establish an explicit form of the distribution of any fixed number of terms uniformly distributed on $[0, 1].$