Chisquared distribution
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Probability density function 

Cumulative distribution function 

Parameters  degrees of freedom 

Support  
CDF  
Mean  
Median  approximately 
Mode  if 
Variance  
Skewness  
Ex. kurtosis  
Entropy  
MGF  for 
CF 
In probability theory and statistics, the chisquare distribution (also chisquared or distribution) is one of the most widely used theoretical probability distributions in inferential statistics, e.g., in statistical significance tests. It is useful because, under reasonable assumptions, easily calculated quantities can be proven to have distributions that approximate to the chisquare distribution if the null hypothesis is true.
If are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable
is distributed according to the chisquare distribution. This is usually written
The chisquare distribution has one parameter:  a positive integer that specifies the number of degrees of freedom (i.e. the number of )
The chisquare distribution is a special case of the gamma distribution.
The bestknown situations in which the chisquare distribution are used are the common chisquare tests for goodness of fit of an observed distribution to a theoretical one, and of the independence of two criteria of classification of qualitative data. However, many other statistical tests lead to a use of this distribution. One example is Friedman's analysis of variance by ranks.
Characteristics
Probability density function
A probability density function of the chisquare distribution is
where denotes the Gamma function, which takes particular values at the halfintegers.
Cumulative distribution function
Its cumulative distribution function is:
where is the lower incomplete Gamma function and is the regularized Gamma function.
Tables of this distribution — usually in its cumulative form — are widely available and the function is included in many spreadsheets and all statistical packages.
Characteristic function
The characteristic function of the Chisquare distribution is
Properties
The chisquare distribution has numerous applications in inferential statistics, for instance in chisquare tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a regression line via its role in Student's tdistribution. It enters all analysis of variance problems via its role in the Fdistribution, which is the distribution of the ratio of two independent chisquared random variables divided by their respective degrees of freedom.
Normal approximation
If , then as tends to infinity, the distribution of tends to normality. However, the tendency is slow (the skewness is and the kurtosis excess is ) and two transformations are commonly considered, each of which approaches normality faster than itself:
Fisher empirically showed that is approximately normally distributed with mean and unit variance. It is possible to arrive at the same normal approximation result by using moment matching. To see this, consider the mean and the variance of a Chidistributed random variable , which are given by and , where is the Gamma function. The particular ratio of the Gamma functions in has the following series expansion :
When , this ratio can be approximated as follows:
Then, simple moment matching results in the following approximation of : , from which it follows that .
Wilson and Hilferty showed in 1931 that is approximately normally distributed with mean and variance .
The expected value of a random variable having chisquare distribution with degrees of freedom is and the variance is . The median is given approximately by
Note that 2 degrees of freedom lead to an exponential distribution.
Information entropy
The information entropy is given by
where is the Digamma function.
Related distributions
 is an exponential distribution if (with 2 degrees of freedom).
 is a chisquare distribution if for independent that are normally distributed.
 If the have nonzero means, then is drawn from a noncentral chisquare distribution.
 The chisquare distribution is a special case of the gamma distribution, in that .
 is an Fdistribution if where and are independent with their respective degrees of freedom.
 is a chisquare distribution if where are independent and .
 if is chisquare distributed, then is chi distributed.
 in particular, if (chisquare with 2 degrees of freedom), then is Rayleigh distributed.
 if are i.i.d. random variables, then where .
 if , then
Name  Statistic 

chisquare distribution  
noncentral chisquare distribution  
chi distribution  
noncentral chi distribution 