  We propose a new setting for testing properties of distributions while
receiving samples from several distributions, but few samples per distribution.
Given samples from $s$ distributions, $p_1, p_2, \ldots, p_s$, we design
testers for the following problems: (1) Uniformity Testing: Testing whether all
the $p_i$'s are uniform or $\epsilon$-far from being uniform in
$\ell_1$-distance (2) Identity Testing: Testing whether all the $p_i$'s are
equal to an explicitly given distribution $q$ or $\epsilon$-far from $q$ in
$\ell_1$-distance, and (3) Closeness Testing: Testing whether all the $p_i$'s
are equal to a distribution $q$ which we have sample access to, or
$\epsilon$-far from $q$ in $\ell_1$-distance. By assuming an additional natural
condition about the source distributions, we provide sample optimal testers for
all of these problems.
