Noncentral chi distribution
Encyclopedia : N : NO : NON : Noncentral chi distribution
I_(\lambda x)]| cdf =| mean =[\mu\equiv\sqrt}L_^\left(\frac\right)\,]| median =| mode =| variance =[k+\lambda^2-\mu^2\,]| skewness =| kurtosis =| entropy =| mgf =| char = }}
In probability theory and statistics, the noncentral chi distribution is a generalization of the chi distribution. If [X_i] are k independent, normally distributed random variables with means [\mu_i] and variances [\sigma_i^2], then the statistic
- [Z = \sqrt\right)^2}]
- [\lambda=\sqrt\right)^2}]
Properties
The probability density function is
- [f(x;k,\lambda)=\fracx^k\lambda}} I_(\lambda x)]
The first few raw moments are:
- [\mu^'_1=\sqrt}L_^\left(\frac\right)]
- [\mu^'_2=k+\lambda^2]
- [\mu^'_3=3\sqrt}L_^\left(\frac\right)]
- [\mu^'_4=(k+\lambda^2)^2+2(k+2\lambda^2)]
Related distributions
- If [X] is chi distributed: [X \sim \chi_k] then [X] is also non-central chi-square distributed: [X \sim NC\chi_k(0)]
| Name | Statistic |
|---|---|
| chi-square distribution | [\sum_1^k \left(\frac\right)^2] |
| noncentral chi-square distribution | [\sum_1^k \left(\frac\right)^2] |
| chi distribution | [\sqrt\right)^2}] |
| noncentral chi distribution | [\sqrt\right)^2}] |
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