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Gamma Distribution Calculator

Work out gamma distribution instantly with clear inputs, formula shown and shareable results.

About Gamma Distribution Calculator

The gamma distribution generalises the exponential to k sequential events, with mean kθ and variance kθ². Its CDF is the regularised lower incomplete gamma function P(k, x/θ).

Formulas

Gamma density

f(x) = x^(k−1)·e^(−x/θ) / (Γ(k)·θᵏ)

Mean and variance

E = kθ, Var = kθ²

Worked example

With the default values below, Gamma Distribution Calculator returns P(X ≤ x) of 0.4561868841. Change any input above and every figure here is recalculated in your browser.

Inputs used

Shape kminimum 0
3
Scale θminimum 0
2
Evaluate at xminimum 0
5

Results

P(X ≤ x)
0.4561868841
Probability density at x
0.1282578103
Mean kθ
6.000000
Variance kθ²
12.000000

Step by step

  1. 1Values used
  2. 2Gamma density
  3. 3Mean and variance
  4. 4P(X ≤ x)= 0.4561868841
  5. 5Probability density at x= 0.1282578103
  6. 6Mean kθ= 6.000000
  7. 7Variance kθ²= 12.000000

Assumptions and limits

  • The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For k = 3, θ = 2 at x = 5?

P(X ≤ 5) ≈ 0.4562, with a mean of 6 and variance of 12.

How does it relate to other distributions?

k = 1 gives the exponential, and k = df/2 with θ = 2 gives the chi-square distribution.

Related Calculators

Gamma Distribution Calculator – Calculate P(X ≤ x)