Chi Probability Distribution Family

See: Chi-Squared Distribution, Normal Distribution, Rayleigh Distribution, Maxwell Distribution.

References

(Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/chi_distribution Retrieved:2016-5-17.
- In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the square root of the sum of squares of independent random variables having a standard normal distribution, or equivalently, the distribution of the Euclidean distance of the random variables from the origin. The most familiar examples are the Rayleigh distribution with chi distribution with 2 degrees of freedom, and the Maxwell distribution of (normalized) molecular speeds which is a chi distribution with 3 degrees of freedom (one for each spatial coordinate). If [math]\displaystyle{ X_i }[/math] are k independent, normally distributed random variables with means [math]\displaystyle{ \mu_i }[/math] and standard deviations [math]\displaystyle{ \sigma_i }[/math], then the statistic : [math]\displaystyle{ Y = \sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2} }[/math] is distributed according to the chi distribution. Accordingly, dividing by the mean of the chi distribution (scaled by the square root of n − 1) yields the correction factor in the unbiased estimation of the standard deviation of the normal distribution. The chi distribution has one parameter: [math]\displaystyle{ k }[/math] which specifies the number of degrees of freedom (i.e. the number of [math]\displaystyle{ X_i }[/math] ).