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. Assuming that , on Chart 1 we see that Hence, using additivity of probability,
The last transition is based on the assumption that for all , which is maintained for continuous random variables throughout the guide by Abdey.
Question 2. What is the link between the densities of and By the Leibniz integral rule (1) implies
Exercise. Assuming that is an increasing differentiable function with the inverse and answer questions similar to 1 and 2.
See the definition of Just applying (2) to and we get
Since the procedure for identifying the gamma distribution gives
We have derived the density of the chi-squared variable with one degree of freedom, see also Example 3.52, J. Abdey, Guide ST2133.
For with independent standard normals we can write where the chi-squared variables on the right are independent and all have one degree of freedom. This is because deterministic (here quadratic) functions of independent variables are independent.