Chi-squared distribution
This post is intended to close a gap in J. Abdey's guide ST2133, which is absence of distributions widely used in Econometrics.
Chi-squared with one degree of freedom
Let be a random variable and let
Question 1. What is the link between the distribution functions of and
The start is simple: just follow the definitions.
(1)
The last transition is based on the assumption that
Question 2. What is the link between the densities of
(2)
Exercise. Assuming that
See the definition of
Since
We have derived the density of the chi-squared variable with one degree of freedom, see also Example 3.52, J. Abdey, Guide ST2133.
General chi-squared
For
Recall that the gamma density is closed under convolutions with the same
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