![]() ![]() If the mean is not equal to zero, a more general distribution known as the noncentral chi. where is a confluent hypergeometric function of the second kind, giving the first few as. ![]() The th raw moment for a distribution with degrees of freedom is. ![]() An interesting aspect of the confidence intervals that we obtained was that they often did not depend on the details of the distribution from which we obtained the random sample. The chi-squared distribution is implemented in the Wolfram Language as ChiSquareDistribution n. Gamma Distribution The gamma distribution is a two-parameter continuous distribution that has parameters a (shape) and b (scale). The chi-square distribution is equivalent to the gamma distribution where k/2 and 2. The F distribution can be defined as the ratio F 1 2 1 2 2 2, where 2 1 and 2 2 are both chi-square distributed with 1 and 2 degrees of freedom, respectively. For upper-tail one-sided tests, the test statistic is compared with a value from the table. In the above discussion, we assumed $n$ to be large so that we could use the CLT. Definition 1: The chi-square distribution with k degrees of freedom, abbreviated 2 (k), has the probability density function (pdf) k does not have to be an integer and can be any positive real number. A test statistic with degrees of freedom is computed from the data. ![]()
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