1.
Exploratory Data Analysis
1.3. EDA Techniques 1.3.6. Probability Distributions 1.3.6.6. Gallery of Distributions


Probability Density Function 
The chisquare distribution results when ν independent variables
with standard normal distributions are squared
and summed. The formula for the probability
density function of the chisquare distribution is
\( f(x) = \frac{e^{\frac{x} {2}}x^{\frac{\nu} {2}  1}} {2^{\frac{\nu} {2}}\Gamma(\frac{\nu} {2}) } \;\;\;\;\;\;\; \mbox{for} \; x \ge 0 \) where ν is the shape parameter and Γ is the gamma function. The formula for the gamma function is \( \Gamma(a) = \int_{0}^{\infty} {t^{a1}e^{t}dt} \) In a testing context, the chisquare distribution is treated as a "standardized distribution" (i.e., no location or scale parameters). However, in a distributional modeling context (as with other probability distributions), the chisquare distribution itself can be transformed with a location parameter, μ, and a scale parameter, σ. The following is the plot of the chisquare probability density function for 4 different values of the shape parameter.


Cumulative Distribution Function 
The formula for the cumulative distribution
function of the chisquare distribution is
\( F(x) = \frac{\gamma(\frac{\nu} {2},\frac{x} {2})} {\Gamma(\frac{\nu} {2})} \;\;\;\;\;\;\; \mbox{for} \; x \ge 0 \) where Γ is the gamma function defined above and γ is the incomplete gamma function. The formula for the incomplete gamma function is \( \Gamma_{x}(a) = \int_{0}^{x} {t^{a1}e^{t}dt} \) The following is the plot of the chisquare cumulative distribution function with the same values of ν as the pdf plots above.


Percent Point Function 
The formula for the percent point
function of the chisquare distribution does not exist in
a simple closed form. It is computed numerically.
The following is the plot of the chisquare percent point function with the same values of ν as the pdf plots above.


Other Probability Functions  Since the chisquare distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit the formulas and plots for the hazard, cumulative hazard, survival, and inverse survival probability functions.  
Common Statistics 


Parameter Estimation  Since the chisquare distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit any discussion of parameter estimation.  
Comments  The chisquare distribution is used in many cases for the critical regions for hypothesis tests and in determining confidence intervals. Two common examples are the chisquare test for independence in an RxC contingency table and the chisquare test to determine if the standard deviation of a population is equal to a prespecified value.  
Software  Most general purpose statistical software programs support at least some of the probability functions for the chisquare distribution. 