distributions

# distributions - is symmetric about zero and tends...

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1 Relations Between the Normal, χ 2 , t and F Distributions Let z N (0 , 1) be a standard normal variable. If n random values z 1 ,z 2 ,...,z n are drawn from this distribution, squared, and summed, the resultant statistic is said to be a χ 2 distribution with n degrees of freedom, ( z 2 1 + z 2 2 + ··· + z 2 n ) χ 2 ( n ) The precise mathematical form of the χ 2 distribution need not worry us here. Rather, the important point is that it constitutes a one-parameter family of distributions, which is conventionally labelled the degrees of freedom (d.f) of the distribution. As the d.f tend to inﬁnity, the χ 2 distribution approaches the normal distribution. Critical values of the χ 2 distribution can be found in statistical tables. The t distribution may be deﬁned in terms of the normal and an independent χ 2 variable. Let z N (0 , 1) and υ χ 2 ( ν ) where z and υ are independently distributed. Then t = z ν υ (1) has Student’s t distribution with ν d.f. The t distribution, like χ 2 , is a one-parameter family. It
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Unformatted text preview: is symmetric about zero and tends asymptotically to the standard normal distribution. Its critical values can be found in statistical tables. The F distribution is deﬁned in terms of two independent χ 2 variables. Let u and υ be in-dependently distributed χ 2 variables with ν 1 and ν 2 degrees of freedom, respectively. Then the statistic F = u/ν 1 υ/ν 2 (2) has the F distribution with ( ν 1 ,ν 2 ) degrees of freedom (critical values can be found in statistical tables). If we square the expression for t , the result may be written as t 2 = z 2 / 1 υ/ν 2 (3) where z 2 , being the square of a standard normal variable, has the χ 2 (1) distribution. Thus t 2 = F (1 ,ν ), that is, the square of a t variable with ν degrees of freedom is an F distribution with (1 .ν ) degrees of freedom. 1...
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