The author considers the probability measure \(M_{n}(k;\alpha)\) on \(\mathbb R\) for \(\alpha\in\mathbb R^{n}\), \(k\in (\mathbb R_{+}^{*})^{n}\) which is related with the Dirichlet distribution \(D_{n}^{(k)}\). The author establishes a Markov-Krein type formula, where \(\alpha=k_{1}+...k_{n}\), which generalizes results given by other authors for special cases of \(k_{i}\) and \(\alpha\). Also \(M_{n}(k;\alpha)\) is obtained as a convolution of special forms of distributions, and, finally, the moments of the measure \(M_{n}(k;\alpha)\) are computed.