  The exact expression is derived for the expected value, $< {p_i}> $, for the
parameter for any bin $i$ of a histogram following a multinomial distribution
derived by sorting $N$ observations into bins of $B$ classes, if $n_i$ of the
observations are found to be sorted into bin $i$. This expected value is found
to be $ < {p_i}> = \frac {n_i + 1} {N + B}$. The expected value for the
variance is found to be $\frac{< p_i > (1-< p_i >)}{N+B+1}$. A general
expression is derived to determine $< {p_i}^z > $ for arbitrary values of $B$
and $z$. These expressions hold provided there is no \emph{a priori} reason for
$p_i$ associated with any bin to have a value that is exactly equal to 0. For
the particular case of the binomial distribution (B=2), these estimators are
tested by examining how often the value of $p_{true}$, the value which is used
to generate sets of pseudo-random binomial variates, falls within 1.96
estimated standard deviations of the estimated value $< p >$. When compared
with the results of identical, earlier reported tests for small sample sizes,
the unbiased estimators derived here predictably outperform
\emph{asymptotically} unbiased estimators
