Gamma Distribution Calculator

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Parameters

Shape Parameter (α)

Scale Parameter (θ)

Calculation Type

Lower bound (x1)

Upper bound (x2)

Distribution Chart

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Gamma Distribution: Definition, Formula, and Applications

Gamma Distribution

Definition: The gamma distribution is a continuous probability distribution that arises naturally in processes for which the waiting times between events are relevant.

Formula:The probability density function (PDF) is given by:

f(x; \alpha, \theta) = \frac{x^{\alpha-1} e^{-x/\theta}}{\theta^\alpha \Gamma(\alpha)}, \quad x > 0

Where:

\Gamma(\alpha) = \int_0^\infty t^{\alpha-1} e^{-t} dt

Where:

$\alpha$ is the shape parameter (determines the basic shape)
$\theta$ is the scale parameter (stretches/compresses the distribution)
$\Gamma(\alpha)$ is the gamma function

Properties

Mean: $E(X) = \alpha\theta$
Variance: $\text{Var}(X) = \alpha\theta^2$
Mode: $(\alpha-1)\theta$ for $\alpha \geq 1$
Support: $(0, \infty)$
Special cases:
- When $\alpha = 1$ , reduces to exponential distribution
- When $\alpha = n/2, \theta = 2$ , becomes chi-square distribution with n degrees of freedom
- For large $\alpha$ , approaches normal distribution

Gamma Distribution Calculator

Calculator

Parameters

Distribution Chart

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Gamma Distribution: Definition, Formula, and Applications

Gamma Distribution

Properties

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