A note on equal distributions

Summary

It is known that the set

$$\left\{ {E\left( {X_{k_n ,n} } \right)\left| {n = 1,2, \cdots } \right.} \right\}, where 1 \leqq k_n \leqq n,$$

of expectations of order statistics of samples from a distributionF which has a finite expectation determinesF. In this note, we show that each of the sets

$$\begin{gathered} \left\{ {E\left( {X_{k_j ,n_j } } \right)\left| {j = 1,2, \cdots } \right.} \right\}, \hfill \\ where \left\{ {\left( {{{k_j } \mathord{\left/ {\vphantom {{k_j } {n_j }}} \right. \kern-\nulldelimiterspace} {n_j }}} \right)\left| {j = 1,2, \ldots } \right.} \right\} is dense in \left[ {0,1} \right], \hfill \\ \end{gathered} $$

$$\begin{gathered} \left\{ {E\left( {X_{1,1} } \right)} \right\} \cup \left\{ {E\left( {X_{k_j ,2j + 1} } \right)\left| {j = 1,2, \cdots } \right.} \right\} \cup \hfill \\ \left\{ {E\left( {X_{k'j,2j + 1} } \right)\left| {j = 1,2, \cdots } \right.} \right\},where 1 \leqq k_j< k'_j \leqq 2j + 1, \hfill \\ \end{gathered} $$

also determinesF.