Mathematica 9 is now available
THIS IS DOCUMENTATION FOR AN OBSOLETE PRODUCT.
SEE THE DOCUMENTATION CENTER FOR THE LATEST INFORMATION.
Wolfram Mathematica Documentation Center
DOCUMENTATION CENTER SEARCH
New to Mathematica? Find your learning path »
Mathematica > Data Manipulation > Statistical Data Analysis > Probability & Statistics > Parametric Statistical Distributions > Normal and Related Distributions > InverseChiSquareDistribution >
Mathematica > Mathematics and Algorithms > Statistical Data Analysis > Probability & Statistics > Parametric Statistical Distributions > Normal and Related Distributions > InverseChiSquareDistribution >
BUILT-IN MATHEMATICA SYMBOLSee Also »|More About »

InverseChiSquareDistribution

InverseChiSquareDistribution[]
represents an inverse distribution with degrees of freedom.
InverseChiSquareDistribution
represents a scaled inverse distribution with degrees of freedom and scale .
MORE INFORMATION
  • The inverse distribution InverseChiSquareDistribution[] is the distribution followed by the inverse of a -distributed random variable with degrees of freedom.
  • The inverse distribution is commonly used in normal models for Bayesian data analysis.
EXAMPLESCLOSE ALL
Basic Examples   (4)
Probability density function:
For scaled inverse distribution:
Cumulative distribution function:
For scaled inverse distribution:
Mean and variance:
Median:
Probability density function:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
For scaled inverse distribution:
In[3]:=
Click for copyable input
Out[3]=
In[4]:=
Click for copyable input
Out[4]=
In[5]:=
Click for copyable input
Out[5]=
 
Cumulative distribution function:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
For scaled inverse distribution:
In[3]:=
Click for copyable input
Out[3]=
In[4]:=
Click for copyable input
Out[4]=
In[5]:=
Click for copyable input
Out[5]=
 
Mean and variance:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
 
Median:
In[1]:=
Click for copyable input
Out[1]=
Scope   (7)
Generate a set of pseudorandom numbers that are inverse distributed:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare a density histogram of the sample with the PDF of the estimated distribution:
Skewness depends only on the number of degrees of freedom:
In the limit the distribution becomes symmetric:
Kurtosis depends only on the number of degrees of freedom:
In the limit kurtosis is the same as for NormalDistribution:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Closed form for symbolic order:
Hazard function:
Quantile function:
For scaled inverse distribution:
Applications   (2)
The posterior distribution of variance of a normal distribution with zero mean was found to be InverseChiSquareDistribution with parameters and scale . Find the most likely value of the variance:
Find the expected variance:
InverseChiSquareDistribution is a conjugate prior for the likelihood of normal distribution with known mean and unknown variance:
Update prior using data sample:
Posterior distribution is again InverseChiSquareDistribution with new parameters and :
Properties & Relations   (8)
Parameter influence on the CDF for each :
InverseChiSquareDistribution is closed under scaling by a positive factor:
Relationships to other distributions:
The two forms are related by a change of variable:
InverseChiSquareDistribution and ChiSquareDistribution have an inverse relationship:
InverseChiSquareDistribution is a special case of PearsonDistribution of type 5:
Possible Issues   (2)
InverseChiSquareDistribution is not defined when either or is not a positive real number:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
New in 7
Ask a question about this page  |  Suggest an improvement  |  Leave a message for the team
Format:   HTML  |  CDF