  We consider the problem of learning an unknown product distribution $X$ over
$\{0,1\}^n$ using samples $f(X)$ where $f$ is a \emph{known} transformation
function. Each choice of a transformation function $f$ specifies a learning
problem in this framework.
  Information-theoretic arguments show that for every transformation function
$f$ the corresponding learning problem can be solved to accuracy $\eps$, using
$\tilde{O}(n/\eps^2)$ examples, by a generic algorithm whose running time may
be exponential in $n.$ We show that this learning problem can be
computationally intractable even for constant $\eps$ and rather simple
transformation functions. Moreover, the above sample complexity bound is nearly
optimal for the general problem, as we give a simple explicit linear
transformation function $f(x)=w \cdot x$ with integer weights $w_i \leq n$ and
prove that the corresponding learning problem requires $\Omega(n)$ samples.
  As our main positive result we give a highly efficient algorithm for learning
a sum of independent unknown Bernoulli random variables, corresponding to the
transformation function $f(x)= \sum_{i=1}^n x_i$. Our algorithm learns to
$\eps$-accuracy in poly$(n)$ time, using a surprising poly$(1/\eps)$ number of
samples that is independent of $n.$ We also give an efficient algorithm that
uses $\log n \cdot \poly(1/\eps)$ samples but has running time that is only
$\poly(\log n, 1/\eps).$
