This article is about handbook of beta distribution and its applications pdf mathematics of the chi-squared distribution. The chi-squared distribution is used primarily in hypothesis testing.
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It arises in the following hypothesis tests, among others. The primary reason that the chi-squared distribution is used extensively in hypothesis testing is its relationship to the normal distribution. Testing hypotheses using a normal distribution is well understood and relatively easy. The simplest chi-squared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-squared distribution could be used. Define a new random variable Q. To generate a random sample from Q, take a sample from Z and square the value.
The subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. However, the normal and chi-squared approximations are only valid asymptotically. For this reason, it is preferable to use the t distribution rather than the normal approximation or the chi-squared approximation for small sample size. De Moivre and Laplace established that a binomial distribution could be approximated by a normal distribution. Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by the normal or the chi-squared distribution.
However, many problems involve more than the two possible outcomes of a binomial, and instead require 3 or more categories, which leads to the multinomial distribution. Just as de Moivre and Laplace sought for and found the normal approximation to the binomial, Pearson sought for and found a multivariate normal approximation to the multinomial distribution. Further properties of the chi-squared distribution can be found in the box at the upper right corner of this article. It follows from the definition of the chi-squared distribution that the sum of independent chi-squared variables is also chi-squared distributed.
Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Unsourced material may be challenged and removed. The chi-squared distribution is also naturally related to other distributions arising from the Gaussian. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables. Several such distributions are described below.
Following are some of the most common situations in which the chi-squared distribution arises from a Gaussian-distributed sample. 05 is often used as a cutoff between significant and not-significant results. The idea of a family of “chi-squared distributions”, however, is not due to Pearson but arose as a further development due to Fisher in the 1920s. Chi-Squared Distributions including Chi and Rayleigh”.
Boca Raton, FL: CRC Press. Evaluating the Normal Approximation to the Binomial Test”. New York: Springer, 2002, eq. The Statistical Analysis of Variance-Heterogeneity and the Logarithmic Transformation”. Yates, Statistical Tables for Biological Agricultural and Medical Research, 6th ed. Two values have been corrected, 7.
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