  We develop differentially private methods for estimating various
distributional properties. Given a sample from a discrete distribution $p$,
some functional $f$, and accuracy and privacy parameters $\alpha$ and
$\varepsilon$, the goal is to estimate $f(p)$ up to accuracy $\alpha$, while
maintaining $\varepsilon$-differential privacy of the sample.
  We prove almost-tight bounds on the sample size required for this problem for
several functionals of interest, including support size, support coverage, and
entropy. We show that the cost of privacy is negligible in a variety of
settings, both theoretically and experimentally. Our methods are based on a
sensitivity analysis of several state-of-the-art methods for estimating these
properties with sublinear sample complexities.
