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The chi-squared distribution (chi-square or ${X^2}$ – distribution) with degrees of freedom, k is the distribution of a sum of the squares of k independent standard normal random variables. It is one of the most widely used probability distributions in statistics. It is a special case of the gamma distribution.
Chi-squared distribution is widely used by statisticians to compute the following:
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Estimation of Confidence interval for a population standard deviation of a normal distribution using a sample standard deviation.
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To check independence of two criteria of classification of multiple qualitative variables.
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To check the relationships between categorical variables.
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To study the sample variance where the underlying distribution is normal.
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To test deviations of differences between expected and observed frequencies.
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To conduct a The chi-square test (a goodness of fit test).
Probability density function
Probability density function of Chi-Square distribution is given as:
Formula
$ begin {cases}
frac{x^{ frac{k}{2} – 1} e^{-frac{x}{2}}}{2^{frac{k}{2}}Gamma(frac{k}{2})}, & text{if $x gt 0 $} \[7pt]
0, & text{if $x le 0 $}
end{cases} $
Where −
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${Gamma(frac{k}{2})}$ = Gamma function having closed form values for integer parameter k.
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${x}$ = random variable.
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${k}$ = integer parameter.
Cumulative distribution function
Cumulative distribution function of Chi-Square distribution is given as:
Formula
= P (frac{x}{2}, frac{k}{2}) }$
Where −
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${gamma(s,t)}$ = lower incomplete gamma function.
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${P(s,t)}$ = regularized gamma function.
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${x}$ = random variable.
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${k}$ = integer parameter.
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