  We approximate the distribution of the sum of independent but not necessarily
identically distributed Bernoulli random variables using a shifted binomial
distribution where the three parameters (the number of trials, the probability
of success, and the shift amount) are chosen to match up the first three
moments of the two distributions. We give a bound on the approximation error in
terms of the total variation metric using Stein's method. A numerical study is
discussed that shows shifted binomial approximations typically are more
accurate than Poisson or standard binomial approximations. The application of
the approximation to solving a problem arising in Bayesian hierarchical
modeling is also discussed.
