  Let $(X_1,X_2,...)$ be a random partition of the unit interval $[0,1]$, i.e.
$X_i\geq0$ and $\sum_{i\geq1} X_i=1$, and let
$(\varepsilon_1,\varepsilon_2,...)$ be i.i.d. Bernoulli random variables of
parameter $p \in (0,1)$. The Bernoulli convolution of the partition is the
random variable $Z =\sum_{i\geq1} \varepsilon_i X_i$. The question addressed in
this article is: Knowing the distribution of $Z$ for some fixed $p\in(0,1)$,
what can we infer about the random partition? We consider random partitions
formed by residual allocation and prove that their distributions are fully
characterised by their Bernoulli convolution if and only if the parameter $p$
is not equal to $1/2$.
