Calculate probability density, cumulative probability, and statistics for the Beta distribution.
Last updated: April 2026
Mean (E[X])
0.2857
Variance
0.0255
Mode
0.2000
PDF(x)
2.160900
CDF(x)
1.038670
| α, β | Shape Description | Mean | Use Case |
|---|---|---|---|
| (1, 1) | Uniform distribution | 0.50 | No prior information |
| (2, 2) | Bell-shaped, symmetric | 0.50 | Weak symmetric belief |
| (2, 5) | Left-skewed (peak left) | 0.29 | Bias toward lower values |
| (5, 2) | Right-skewed (peak right) | 0.71 | Bias toward higher values |
| (10, 10) | Narrow peak, very concentrated | 0.50 | Strong confident belief |
Mean = α/(α+β). Larger α values pull the distribution right; larger β values pull it left. Larger total (α+β) concentrates the distribution.
The Beta distribution is a continuous probability distribution defined on the interval [0, 1], making it ideal for modeling probabilities, proportions, and rates. It's highly flexible—by adjusting parameters α and β, it can represent many different distribution shapes.
Parameters: α (alpha) and β (beta) determine the shape. Both must be positive. The distribution is symmetric when α = β and is used extensively in Bayesian statistics for modeling prior and posterior beliefs about proportions.
Scenario: You flip a coin 7 times and get 2 heads, 5 tails. What's your best estimate of the true probability of heads?
Using Beta(2, 5): The parameters represent prior observations (weak belief) plus the data you observed.
This is a Bayesian posterior distribution. With more flips, α and β would increase, concentration would tighten, and uncertainty would decrease.
How does Beta relate to Bayesian statistics?
The Beta distribution is the conjugate prior for binomial data. If you start with a beta prior and observe successes/failures, the posterior is also beta—making updates simple.
What do α and β control?
Both control shape. α > β means right-skewed (bias toward 1), α < β means left-skewed (bias toward 0), α = β means symmetric. Larger total (α+β) creates sharper peaks.
When is the mode undefined?
When α ≤ 1 or β ≤ 1, there's no interior mode. Beta(0.5, 0.5) is U-shaped; Beta(1, 1) is flat; only α,β > 1 have a peak inside [0, 1].
Why bounded to [0, 1]?
Perfect for probabilities, proportions, and percentages—all naturally lie between 0% and 100%. You can scale to other ranges if needed.
How to fit Beta to real data?
Use method of moments: compute sample mean and variance, then solve for α and β. Or use maximum likelihood estimation (MLE) for more precision.
What about Beta(1, 1)?
Beta(1, 1) is uniform—no preference. Great when you have no prior information. It's equivalent to a neutral starting point in Bayesian analysis.
Can I use Beta for non-probability values?
Yes! Transform any [c, d] interval to [0, 1]: X' = (X - c)/(d - c). Then use Beta. The scaled beta distribution preserves shape properties.
How do PDFs and CDFs differ?
PDF is the probability density at x (height of the curve). CDF is the cumulative probability up to x (area under the curve from 0 to x).
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