Noncentral Chi-Square Distribution Calculator

Compute the PDF, CDF, upper tail probability, mean, and variance of the noncentral chi-square distribution for degrees of freedom k and noncentrality lambda at a value x.

Input

For the noncentral chi-square distribution with degrees of freedom k and noncentrality lambda, compute the probability density, cumulative distribution, and upper tail probability at a value x.

x (a value of 0 or more)

Enter the value x at which to evaluate the probability.

Degrees of freedom k

Enter a positive value for the degrees of freedom k (non-integers allowed).

Noncentrality lambda

Enter a noncentrality lambda of 0 or more. A value of 0 gives the central chi-square distribution.

Result

Lower tail probability F(x = 8) (df 3, noncentrality 2)

0.81759729

Upper tail probability 1 − F(x)

0.18240271

Probability density f(x)

0.05187103

Mean k + lambda

Variance

Probability density function PDF graph

Cumulative distribution function CDF graph

How it works

The noncentral chi-square distribution is the distribution of the sum of squares of independent normal variables with nonzero means. It has two parameters: degrees of freedom k and noncentrality lambda. When lambda equals 0 it reduces to the ordinary central chi-square distribution.
The cumulative distribution is computed as F(x) = sum over j of e^(−lambda/2)(lambda/2)^j / j! times P(k/2+j, x/2), where P(s, t) is the regularized lower incomplete gamma function. This is a Poisson-weighted mixture of central chi-square distributions.
The probability density mixes central chi-square densities with the same Poisson weights: f(x) = sum over j of e^(−lambda/2)(lambda/2)^j / j! times g(x; k+2j), where g is the central chi-square density with k+2j degrees of freedom.
The mean equals k+lambda and the variance equals 2(k+2 lambda). Larger noncentrality lambda shifts the distribution to the right and widens its tail.
The upper tail probability is 1−F(x) and is commonly used in statistical power calculations. This tool evaluates the series outward from the mode of the Poisson weights in both directions for numerical stability.

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