Law of the χ ²

The law of the χ ² (“khi-deux” delivery or “khi square”) is a law with Densité of probability. This law is characterized by a parameter known as degrees of freedom to value in the whole of the natural entireties (nonnull).

That is to say $X_1, \ ldots, X_k$ $k$ random variable independent of the same normal Law centered and reduced, then by definition the variable $X$ , such as

$X: = \ Sigma_ {i=1} ^k X_i^2$

a law of the χ ² follows to K degrees of freedom.

Either $X~$ a random variable according to a law of the χ ² to $k~$ degrees of freedom, one will note $\ chi^2 (K) ~$ the law of $X~$ .

Then the density of $X~$ noted $f_X~$ will be:

$f_X (T) = \ frac {1} {2^ \ frac {K} {2} \ Gamma (\ frac {K} {2})} t^ {\ frac {K} {2} - 1} e^ {- \ frac {T} {2}} \,$ for all T positive

where Γ is the Fonction gamma.

The mathematical Espérance of $X$ is worth $k$ and its Variance is worth $2k$ .

Approximation

When K is “large” ( K > 100), the law of the χ ² can be approximated by a normal law of hope K and variance 2 K .

Use

The principal use of this law consists in appreciating the adequacy of a law of probability to an empirical distribution by using the Test of the χ ² based on the Loi multinomiale. More generally it applies in the test of assumptions to certain thresholds (independence in particular).

Bond with the methods bayésiennes

In its work rational Décisions in dubious the (1974), which constitutes a sum of the technical bayésiennes whose great emergence is made at that time, the professor Myron Tribus watch that the χ ² constitutes an example of passage in extreme cases of the psi-test (test of plausibility) bayésien when the number of involved values becomes large - what is the work condition of the traditional statistics, but not necessarily of the bayésiennes. The connection between the two disciplines, which are asymptotically convergent, is thus complete.

The reference book of Jaynes also gives of it a demonstration on page 287. Bond towards the introduction of this book

Simple: Chi-public garden distribution

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Law of the χ ²

Approximation

Use

Bond with the methods bayésiennes

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