Noncentral chi-square distribution
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} I_(\sqrt)]| cdf =| mean =[k+\lambda\,]| median =| mode =| variance =[2(k+2\lambda)\,]| skewness =[\frac(k+3\lambda)}}]| kurtosis =[\frac]| entropy =| mgf =[\frac\right)}}]| char =[\frac\right)}}] }}
In probability theory and statistics, the noncentral chi-square distribution is a generalization of the chi-square distribution. If [X_i] are k independent, normally distributed random variables with means [\mu_i] and variances [\sigma_i^2], then the random variable
- [Z=\sum_^k \left(\frac\right)^2]
- [\lambda=\sum_1^k \left(\frac\right)^2]
Properties
The probability density function is
- [f(x;k,\lambda)=\fracx^\sqrt}} I_(\sqrt)]
The moment generating function is given by:
- [M(t;k,\lambda)=\frac\right)}}]
- [\mu^'_1=k+\lambda]
- [\mu^'_2=(k+\lambda)^2 + 2(k + 2\lambda) ]
- [\mu^'_3=(k+\lambda)^3 + 6(k+\lambda)(k+2\lambda)+8(k+3\lambda)]
- [\mu^'_4=(k+\lambda)^4 - 12(k+\lambda)^2(k+2\lambda) - 4(11k^2+44k\lambda+36\lambda^2) - 48(k+4\lambda)]
- [\mu_2=2(k+2\lambda)\,]
- [\mu_3=8(k+3\lambda)\,]
- [\mu_4=12(k+2\lambda)^2+48(k+4\lambda)\,]
1. Start with the joint PDF of two independent non-zero mean Gaussian distributions, X and Y
2. Convert the joint density [f(X, Y)] to polar: [f(R, A)] where [R^2 = (X^2+Y^2)], [tan(A) = Y/X]
3. Integrate over the angular variable A
4. Convert from R to r where [r^2 = R]
This will yield a series expansion in r one factor of which matches the modified Bessel function [I_0]
5. Take the Laplace (Fourier) transform term-by-term and the the special case K = 2 and the MGF (characteristic function) will result.
6. For the general case, take the K = 2 MGF (char. func.) and raise it to the [K/2] power
7. The final trick to hide the K-dependence in the numerator of the MGF (char. func.) is to note that [\lambda] is a function of K.
That is,
- [\lambda_2=\sum_1^2 \left(\frac\right)^2]
- [\lambda_K=\sum_1^k \left(\frac\right)^2 = \lambda]
Related distributions
- If [X] is chi-square distributed [X \sim \chi_k^2] then [X^2] is also non-central chi-square distributed: [X^2 \sim NC\chi^2_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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