  This article offers a simplified approach to the distribution theory of
randomly weighted averages or $P$-means $M_P(X):= \sum_{j} X_j P_j$, for a
sequence of i.i.d.random variables $X, X_1, X_2, \ldots$, and independent
random weights $P:= (P_j)$ with $P_j \ge 0$ and $\sum_{j} P_j = 1$. The
collection of distributions of $M_P(X)$, indexed by distributions of $X$, is
shown to encode Kingman's partition structure derived from $P$. For instance,
if $X_p$ has Bernoulli$(p)$ distribution on $\{0,1\}$, the $n$th moment of
$M_P(X_p)$ is a polynomial function of $p$ which equals the probability
generating function of the number $K_n$ of distinct values in a sample of size
$n$ from $P$: $E (M_P(X_p))^n = E p^{K_n}$. This elementary identity
illustrates a general moment formula for $P$-means in terms of the partition
structure associated with random samples from $P$, first developed by Diaconis
and Kemperman (1996) and Kerov (1998) in terms of random permutations. As shown
by Tsilevich (1997) if the partition probabilities factorize in a way
characteristic of the generalized Ewens sampling formula with two parameters
$(\alpha,\theta)$, found by Pitman (1992), then the moment formula yields the
Cauchy-Stieltjes transform of an $(\alpha,\theta)$ mean. The analysis of these
random means includes the characterization of $(0,\theta)$-means, known as
Dirichlet means, due to Von Neumann (1941), Watson (1956) and Cifarelli and
Regazzini (1990) and generalizations of L\'evy's arcsine law for the time spent
positive by a Brownian motion, due to Darling (1949) Lamperti (1958) and
Barlow, Pitman and Yor (1989).
