Gamma Distribution Percent Point
Find the percent point (quantile) x of the gamma distribution from probability p, tail, shape k, and scale theta using the inverse regularized incomplete gamma. Lower and upper tails supported.
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Input
Compute the percent point (quantile, inverse CDF) of the gamma distribution. Enter the probability, tail, shape k, and scale theta.
A probability between 0 and 1 (e.g. 0.95)
Whether p refers to the lower or upper tail probability
A real number greater than 0
A real number greater than 0
Result
Percent point x for p = 0.95 (Lower tail F(x) = p)
4.74386452
Shape k
2
Scale theta
1
Lower probability F(x)
0.95
Upper probability 1 minus F(x)
0.05
Density f(x)
0.04129506
Mean
2
Variance
2
Probability density function PDF
Cumulative distribution function CDF
How it works
- The density is f(x) = x^(k-1) e^(-x/theta) / (theta^k Gamma(k)) and the cumulative distribution is F(x) = P(k, x/theta), the regularized lower incomplete gamma.
- The lower percent point is the x with F(x) = p, while the upper percent point is the x with 1 minus F(x) = p, that is F(x) = 1 minus p.
- The inverse x is obtained by solving P(k, z) = target probability for the standardized variable z = x/theta. The tool combines Newton iteration with bisection for high accuracy.
- The mean is k times theta, the variance is k times theta squared, and the standard deviation is its square root.
- k is the shape parameter (greater than 0), theta is the scale parameter (greater than 0), and p is a probability between 0 and 1.
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Gamma Distribution Percent Point