Gamma and Beta Distribution Tool

Math Probability • Continuous Probability Distributions

Written by STEM Calculators Team Published February 14, 2026 Updated February 24, 2026

Compute PDF, CDF, and interval probabilities for Gamma and Beta distributions. Includes a shaded PDF plot with Play animation.

Distribution

\(\alpha\) (shape)

\(\alpha>0\). Smaller \(\alpha\) can create a spike near 0 for Gamma and a U-shape for Beta.

\(\beta\) (rate for Gamma; shape for Beta)

Gamma uses rate \(\beta\) (so mean is \(\alpha/\beta\)). For Beta, \(\beta\) is a shape parameter.

Probability mode

\(x\)

Gamma: \(x\ge 0\). Beta: \(0\le x\le 1\).

Ready

PDF curve (shaded probability)

Animation ready

The plot shows the PDF. The shaded area equals the probability from the chosen mode. Bound values and the computed probability are shown inside the plot box.

Choose Gamma or Beta, set parameters, then click “Calculate”.

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Theory

What this tool computes

This calculator works with two important continuous distributions: Gamma (on \([0,\infty)\)) and Beta (on \([0,1]\)). For each distribution it computes:

the PDF \(f(x)\) (density),
the CDF \(F(x)=P(X\le x)\),
interval probabilities \(P(a\lt X\lt b)=F(b)-F(a)\),
mean and variance (useful for checking reasonableness).

The graph shows the PDF curve and shades the area corresponding to the chosen probability. The shaded area is the probability (because probability equals “area under the density”).

Gamma distribution \(\Gamma(\alpha,\beta)\) (shape \(\alpha\), rate \(\beta\))

In this tool the Gamma distribution is parameterized by shape \(\alpha>0\) and rate \(\beta>0\) (note: some textbooks use a scale \(\theta=1/\beta\)).

\[ f(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x},\qquad x\ge 0. \]

The Gamma distribution is often used for waiting times: for example, the waiting time until the \(\alpha\)-th event in a Poisson process (with appropriate parameter mapping). Changing \(\alpha\) changes the shape (from highly right-skewed to more bell-like), and changing \(\beta\) (rate) rescales the distribution along the x-axis.

Mean and variance

\[ E[X]=\frac{\alpha}{\beta},\qquad \mathrm{Var}(X)=\frac{\alpha}{\beta^2}. \]

CDF and incomplete gamma

The CDF does not usually have an elementary closed form. It is written using the lower incomplete gamma function \(\gamma(\alpha,z)\) and the Gamma function \(\Gamma(\alpha)\):

\[ F(x)=P(X\le x)=\frac{\gamma(\alpha,\beta x)}{\Gamma(\alpha)} \;=\; P(\alpha,\beta x), \]

where \(P(\alpha,\beta x)\) is the regularized incomplete gamma. Numerically, it can be computed accurately using a series expansion (small \(x\)) or continued fractions (large \(x\)).

Beta distribution \(\mathrm{Beta}(\alpha,\beta)\) on \([0,1]\)

The Beta distribution has two shape parameters \(\alpha>0\), \(\beta>0\) and lives on \([0,1]\). It is flexible: it can be uniform, U-shaped, right-skewed, left-skewed, or mound-shaped depending on \(\alpha,\beta\).

\[ f(x)=\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)},\qquad 0\le x\le 1, \]

Here \(B(\alpha,\beta)\) is the Beta function, related to the Gamma function:

\[ B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}. \]

Mean and variance

\[ E[X]=\frac{\alpha}{\alpha+\beta},\qquad \mathrm{Var}(X)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}. \]

CDF and incomplete beta

The CDF uses the regularized incomplete beta function \(I_x(\alpha,\beta)\):

\[ F(x)=P(X\le x)=I_x(\alpha,\beta). \]

In practice, \(I_x(\alpha,\beta)\) is computed using stable continued fraction methods (with symmetry transforms to improve accuracy).

Interval probabilities and the shaded area on the graph

For any continuous distribution with CDF \(F\),

\[ P(a\lt X\lt b)=\int_a^b f(x)\,dx = F(b)-F(a). \]

That is exactly what the shaded region represents: the area under the PDF between the two vertical bounds. The tool displays the bound values inside the plot, and shows the computed probability inside the plot box as well.

University note: conjugate priors (why Beta/Gamma show up in Bayesian stats)

Beta and Gamma distributions are famous because they act as conjugate priors for common likelihoods:

Beta–Binomial: If \(p\sim \mathrm{Beta}(\alpha,\beta)\) and you observe binomial data, the posterior for \(p\) is also Beta with updated parameters.
Gamma–Poisson: If \(\lambda\sim \Gamma(\alpha,\beta)\) and counts are Poisson, the posterior for \(\lambda\) stays Gamma (parameters update additively).
Gamma–Exponential: For exponential waiting-time models with unknown rate, Gamma is also conjugate.

Intuition: the prior and posterior share the same functional form, so updates become simple parameter changes rather than an entirely new distribution family.

PDF curve (shaded probability)

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