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Evaluate any Beta(alpha, beta) distribution at a point x and read its density, cumulative probability, mean, variance, mode and standard deviation at once.

📘 How to Use

  1. Enter shape parameters alpha and beta (both greater than 0).
  2. Enter the point x where you want the probability, between 0 and 1.
  3. Read the density f(x), cumulative F(x), mean, variance, mode and standard deviation.

Beta Distribution Probability Calculator

Alpha shapes the left tail. In Bayesian priors this is often successes + 1.

Beta shapes the right tail. In Bayesian priors this is often failures + 1.

Point at which to evaluate the distribution, between 0 and 1.

Density f(x)
1.1340
Probability density at the chosen x.
Cumulative F(x) = P(X ≤ x)
0.5798 (57.98%)
Probability that X falls at or below x.
Mean E[X]
0.2857
Variance
0.0255
Mode
0.2000
Std deviation
0.1597

※ f(x) = x^(alpha-1) (1-x)^(beta-1) / B(alpha, beta)

※ Used as the conjugate prior for binomial success rates in Bayesian inference.

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Beta Distribution Probability Calculator | PDF, CDF and 4 Statistics at Once

Enter shape parameters alpha and beta plus an evaluation point x, and get the Beta(alpha, beta) density f(x), cumulative probability F(x), mean, variance, mode and standard deviation in a single panel. Built for Bayesian posteriors and binomial rate analysis.

💡 About this tool

The Beta distribution lives on the interval [0, 1], which makes it the natural model for probabilities, proportions and conversion rates. If you run A/B tests or estimate a click-through rate, you have almost certainly met it as the conjugate prior of the binomial: start from a prior Beta(alpha, beta), observe s successes and f failures, and the posterior is simply Beta(alpha + s, beta + f).

The painful part by hand is the CDF. There is no closed form for the regularized incomplete beta function I_x(alpha, beta), so you either reach for a table or write your own series. This tool evaluates it with a Lanczos log-gamma approximation and the Numerical Recipes continued fraction, then reports six statistics at once so you can sweep alpha and beta and watch the shape shift without re-deriving anything.

🧐 Frequently Asked Questions

Why can the density f(x) be greater than 1? A density is not a probability. Only the area under the curve integrates to 1, so a distribution concentrated in a narrow band can have a peak density well above 1. The value bounded between 0 and 1 is the cumulative F(x).

What do alpha and beta actually control? Alpha pulls probability mass toward 1 and beta pulls it toward 0. Their sum controls concentration: a larger alpha + beta gives a sharper, narrower peak around the mean alpha / (alpha + beta).

How do I read this as a Bayesian posterior? Treat alpha as successes + 1 and beta as failures + 1 from your prior plus observed data. The mean alpha / (alpha + beta) is your point estimate for the underlying rate, and the spread of the curve tells you how sure you can be.

Why does the mode sometimes show "0, 1" or "0-1"? The mode formula (alpha - 1) / (alpha + beta - 2) only gives an interior peak when both alpha and beta exceed 1. When both are below 1 the mass piles at the two endpoints, and when alpha = beta = 1 the distribution is uniform, so the tool flags those cases instead of printing a misleading single number.

What if x falls outside 0 to 1? The Beta distribution has zero density outside [0, 1], so any out-of-range x is clamped to the nearest boundary before the statistics are computed.

📚 Why the Beta distribution shows up everywhere in experimentation

Once you frame a conversion rate as "successes out of trials," the Beta distribution becomes the language of uncertainty around that rate. A button with 8 clicks from 40 impressions is not a flat 20% — it is Beta(9, 33), a curve you can compare directly against a rival variant by checking how much of one posterior sits above the other.

That self-similar update — prior Beta in, posterior Beta out — is what makes it a conjugate prior, and it is why so many sequential testing frameworks lean on it. Every new batch of data just adds to alpha and beta, so the estimate keeps refining without ever leaving the Beta family or needing a fresh integral.