The beta density

Chapter 3 · A distribution over a probability r between 0 and 1, with density p(r) = Γ(α+β)/(Γ(α)Γ(β)) · r^α−1(1−r)^β−1. The two parameters α and β control its shape, and they can be read as imagined heads and tails.

Examples from the book:

━ density ╎ mean α/(α+β)

Mean:

Variance:

Imagined tosses α + β:

α = β = 1 is flat: every r is equally likely. α = β gives a density symmetric about 1/2, narrower as α + β grows (a more confident belief that the coin is fair). α > β leans toward heads, α < β toward tails. With α or β below 1 the density piles up at an end of the interval and rises without limit there.