Draw N points from a Beta distribution and examine how their structure shapes the gaps between them
Beta Distribution
Beta(α, β) is a flexible distribution over [0, 1]
that models uncertainty about an unknown probability — for example,
the bias p of a coin.
🪙Coin interpretation.
Alpha and beta are pseudo-counts — they encode
a prior belief plus observed data. The uniform prior on
[0, 1] is equivalent to assuming you already saw
1 head and 1 tail before collecting any data,
giving the starting point Beta(1, 1).
Additional observations accumulate:
α =1 (prior) + 0 observed heads
β =1 (prior) + 0 observed tails
E[X]
0.5000
α / (α + β)
Var[X]
0.0833
αβ / [(α+β)²(α+β+1)]
Differential Entropy h (bits)
0.000
−∫ f(x) log₂ f(x) dx · can be < 0
extra bits to describe outcomes vs. U(0, 1); negative means more concentrated
Continuous Perplexity
1.000
exp(h)
effective width of a uniform interval with the same information content
α (alpha)1.0
0.1 (not 0: degenerate)50
β (beta)1.0
0.1 (not 0: degenerate)50
N (points)500
1002000
Quick shapes
Sample
N sorted points (order statistics) from Beta(α, β) on [0, 1]
Δx (gaps between adjacent sorted samples): light = small · dark = large (rank-normalized)
Sample histogram · Beta(α, β) PDF
Sample Spacings
Histogram of N+1 gaps between sorted points (including endpoints 0 and 1)