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In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables .
The chi-squared distribution is the maximum entropy probability distribution for a random variate for which = and ( ()) = (/) + are fixed. Since the chi-squared is in the family of gamma distributions, this can be derived by substituting appropriate values in the Expectation of the log moment of gamma .
The Birnbaum–Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times. The chi distribution. The noncentral chi distribution; The chi-squared distribution, which is the sum of the squares of n independent
A gamma distribution with shape parameter α = v/2 and rate parameter β = 1/2 is a chi-squared distribution with ν degrees of freedom. A chi-squared distribution with 2 degrees of freedom (k = 2) is an exponential distribution with a mean value of 2 (rate λ = 1/2 .)
A discrete probability distribution is the probability distribution of a random variable that can ... the distribution of the ratio of two scaled chi squared ...
Here is one based on the distribution with 1 degree of freedom. Suppose that X {\displaystyle X} and Y {\displaystyle Y} are two independent variables satisfying X ∼ χ 1 2 {\displaystyle X\sim \chi _{1}^{2}} and Y ∼ χ 1 2 {\displaystyle Y\sim \chi _{1}^{2}} , so that the probability density functions of X {\displaystyle X} and Y ...
The probability density function (pdf) is given by (;,) = = / (/)! + (),where is distributed as chi-squared with degrees of freedom.. From this representation, the noncentral chi-squared distribution is seen to be a Poisson-weighted mixture of central chi-squared distributions.
Chi-squared distribution, showing χ 2 on the x-axis and p-value (right tail probability) on the y-axis.. A chi-squared test (also chi-square or χ 2 test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large.