Noncentral Chi-Square Poisson Mixture
Resolve a noncentral squared-normal law into ordinary chi-square components.
Free theory articleResolve a noncentral squared-normal law into ordinary chi-square components.
Concept & interpretation
The noncentral chi-square distribution arises when standard normal variables with non-zero means are squared and summed. Remarkably, this complicated distribution can be exactly resolved into an infinite mixture of ordinary (central) chi-square distributions, weighted by Poisson probabilities.
Interactive Experiment
Open the Noncentral Chi-Square Poisson Mixture module.
- Set Degrees of freedom to 2 and Noncentrality to 2.
- Set Probe x to 2 and Mixture index to 1. Notice how the Component density reads approximately 0.183939720586.
- Change the Mixture index to 0. The Poisson component weight jumps to 0.367879441171, but the component degrees drops to 2, and the component density drops to 0.183939720586.
- Adjust Noncentrality to 8. You will see the total noncentral density curve shift dramatically rightward as the mean grows to ν + λ = 10.