0.1282582
6
12
3.464102
4
1.154701
0.57735
-0.288675
0.1282582
6
12
3.464102
4
1.154701
0.57735
-0.288675
The Gamma Distribution Calculator computes the probability density, approximate cumulative probability, and statistics for the gamma distribution. This flexible continuous distribution is defined by a shape parameter $$k$$ and a scale parameter $$\theta$$, and it generalizes the exponential and chi-squared distributions.
The gamma distribution has the probability density function:
$$f(x; k, \theta) = \frac{x^{k-1} e^{-x/\theta}}{\theta^k \Gamma(k)}, \quad x > 0$$
where $$\Gamma(k) = \int_0^\infty t^{k-1} e^{-t}\,dt$$ is the gamma function. For positive integers, $$\Gamma(n) = (n-1)!$$.
The mean is $$E[X] = k\theta$$, and the variance is $$\text{Var}(X) = k\theta^2$$. The mode is $$(k-1)\theta$$ for $$k \geq 1$$ and 0 for $$k < 1$$. The skewness is $$\frac{2}{\sqrt{k}}$$, indicating the distribution is always right-skewed but becomes more symmetric as $$k$$ increases.
The gamma distribution encompasses several important distributions as special cases. When $$k = 1$$, it reduces to the exponential distribution with rate $$1/\theta$$. When $$k = n/2$$ and $$\theta = 2$$, it becomes the chi-squared distribution with $$n$$ degrees of freedom. The Erlang distribution is the gamma with integer shape $$k$$.
A key property: the sum of $$k$$ independent exponential random variables with rate $$1/\theta$$ follows a gamma distribution with shape $$k$$ and scale $$\theta$$. This makes the gamma distribution natural for modeling the total waiting time for $$k$$ events in a Poisson process.
The gamma distribution is used extensively in reliability engineering (time to failure when failures accumulate), insurance (aggregate claim amounts), meteorology (rainfall amounts), Bayesian statistics (conjugate prior for the Poisson rate and precision of normal distributions), and queuing theory (service time distributions).
Enter the shape parameter $$k > 0$$, scale parameter $$\theta > 0$$, and a value $$x \geq 0$$. The calculator computes the PDF, an approximate CDF, mean, variance, standard deviation, mode, and skewness.
The PDF is computed using logarithmic form: $$\ln f(x) = (k-1)\ln x - x/\theta - k\ln\theta - \ln\Gamma(k)$$, then exponentiated. The log-gamma function uses Stirling's approximation. The CDF approximation uses an asymptotic expansion. All statistical moments use exact formulas.
The PDF value indicates relative likelihood at $$x$$. For $$k < 1$$, the PDF approaches infinity as $$x \to 0$$. For $$k = 1$$, the PDF starts at $$1/\theta$$ (exponential shape). For $$k > 1$$, the PDF rises to a peak at the mode $$(k-1)\theta$$, then decays. Larger $$\theta$$ stretches the distribution to the right. Larger $$k$$ makes it more symmetric and bell-shaped.
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For Gamma(3, 2), the mean is 6, mode is 4, and standard deviation is about 3.46. The PDF at x = 5 is approximately 0.104.
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Gamma(1, 5) is the exponential distribution with mean 5. The mode is at 0 (monotonically decreasing PDF), and skewness is 2.
The shape parameter $$k$$ controls the form of the distribution: whether it is monotonically decreasing ($$k < 1$$), exponential ($$k = 1$$), or bell-shaped ($$k > 1$$). The scale parameter $$\theta$$ stretches the distribution horizontally—multiplying $$\theta$$ by 2 doubles the mean, variance, and mode.
The chi-squared distribution with $$n$$ degrees of freedom is a gamma distribution with shape $$k = n/2$$ and scale $$\theta = 2$$. This connection is fundamental in statistical hypothesis testing.
Some references use rate $$\beta = 1/\theta$$ instead of scale $$\theta$$. The PDF becomes $$f(x) = \frac{\beta^k x^{k-1} e^{-\beta x}}{\Gamma(k)}$$. The mean becomes $$k/\beta$$ and the variance $$k/\beta^2$$. Both parameterizations describe the same family.
Since $$x \geq 0$$ and the distribution has a long right tail, it is always right-skewed. The skewness $$2/\sqrt{k}$$ decreases as $$k$$ increases, and by the central limit theorem the gamma approaches normality for large $$k$$ (it is a sum of $$k$$ exponentials).
The gamma distribution is the conjugate prior for the Poisson rate parameter $$\lambda$$ and for the precision (inverse variance) of a normal distribution. This conjugacy makes posterior computation straightforward in Bayesian models.
Yes. Individual claim amounts are often modeled with gamma distributions because they are positive, right-skewed, and have a flexible shape. The aggregate claims (sum) also follow a gamma distribution under certain assumptions, making it a cornerstone of actuarial science.
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