Two Monotonicity Results for Beta Distribution Functions

Write pbeta(·,α,β) for the distribution function of the Beta distribution with parameters α and β. We show that α↦pbeta(α/(α+β),α,β) is decreasing and α↦pbeta(α/(α+β),α+1,β) is increasing over the positive reals, with the common limit for α→∞ expressible in terms of the Gamma distribution functions, and discuss implications for the distribution functions of the Gamma, Poisson and Binomial distributions.