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ChiSquareDistribution

ChiSquareDistribution[]
represents a distribution with degrees of freedom.
  • The probability density for value in a distribution is proportional to for , and is zero for . »
  • For integer , the distribution with degrees of freedom gives the distribution of sums of squares of values sampled from a normal distribution.
Probability density function:
Cumulative distribution function:
Mean and variance:
Median:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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Median:
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Generate a set of pseudorandom numbers that are distributed:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare density histogram of the sample with the PDF of the estimated distribution:
Skewness:
For a large number of degrees of freedom the distribution becomes symmetric:
Kurtosis:
The limiting value is the kurtosis of NormalDistribution:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Closed form for symbolic order:
Cumulant has closed form:
Hazard function:
Quantile function:
ChiSquareDistribution is used in exact (small) sampling theory. Define statistics:
If data comes from a NormalDistribution, then statistics follow ChiSquareDistribution, even for data that is a sample of small size (less than 30):
The weight, in grams, of a particular boxed cereal product is known to follow a normal distribution. A quality assurance team samples 15 boxes at random and records their weights. Test the hypothesis that the standard deviation product weight is less than 36:
Under the null hypothesis of the following statistic follows ChiSquareDistribution:
The null hypothesis cannot be rejected at the 5% level:
Assuming the product weight standard deviation equals 32, compute the probability of rejecting the null hypothesis, also known as the power of the test, at the 5% level as a function of sample size:
Find the sample size required for the power of the test to be at least 80%:
Parameter influence on the CDF for each :
ChiSquareDistribution[] converges to a normal distribution as ->∞:
Sum of -distributed variables follows distribution:
Relationships to other distributions:
NoncentralChiSquareDistribution simplifies to distribution:
distribution is a limiting case of FRatioDistribution:
The ratio of two -distributed variables follows FRatioDistribution:
Sum of squares of variables from NormalDistribution follows distribution:
distribution is a special case of GammaDistribution:
The square root of a variable follows the ChiDistribution:
Square of RayleighDistribution with is a special case of distribution:
Square of MaxwellDistribution with is a special case of distribution:
distribution and InverseChiSquareDistribution have an inverse relationship:
distribution is a special case of type 3 PearsonDistribution:
A transformation of distribution yields BetaDistribution:
is a transformation of UniformDistribution:
distribution is a transformation of LaplaceDistribution:
For sum of variables:
distribution is a transformation of ParetoDistribution:
distribution is a transformation of ParetoDistribution:
StudentTDistribution is a transformation of distribution:
ChiSquareDistribution is not defined when is not a positive real number:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
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