28
Dec 21

Chi-squared distribution

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 X be a random variable and let Y=X^{2}.

Question 1. What is the link between the distribution functions of Y and X?

Chart 1. Inverting a square function

Chart 1. Inverting a square function

The start is simple: just follow the definitions. F_{Y}\left( y\right)=P\left( Y\leq y\right) =P\left( X^{2}\leq y\right) . Assuming that y>0, on Chart 1 we see that \left\{ x:x^{2}\leq y\right\} =\left\{x: -\sqrt{y}\leq x\leq \sqrt{y}\right\} . Hence, using additivity of probability,

(1) F_{Y}\left( y\right) =P\left( -\sqrt{y}\leq X\leq \sqrt{y}\right)  =P\left( X\leq \sqrt{y}\right) -P\left( X<-\sqrt{y}\right)

=F_{X}\left( \sqrt{y}\right) -F_{X}\left( -\sqrt{y}\right) .

The last transition is based on the assumption that P\left( X<x  \right) =P\left( X\leq x\right) , for all x, which is maintained for continuous random variables throughout the guide by Abdey.

Question 2. What is the link between the densities of X and Y=X^{2}? By the Leibniz integral rule (1) implies

(2) f_{Y}\left( y\right) =f_{X}\left( \sqrt{y}\right) \frac{1}{2\sqrt{y}}  +f_{X}\left( -\sqrt{y}\right) \frac{1}{2\sqrt{y}}.

Exercise. Assuming that g is an increasing differentiable function with the inverse h and Y=g(X) answer questions similar to 1 and 2.

See the definition of \chi _{1}^{2}. Just applying (2) to X=z and   Y=z^{2}=\chi _{1}^{2} we get

f_{\chi _{1}^{2}}\left( y\right) =\frac{1}{\sqrt{2\pi }}e^{-y/2}\frac{1}{2  \sqrt{y}}+\frac{1}{\sqrt{2\pi }}e^{-y/2}\frac{1}{2\sqrt{y}}=\frac{1}{\sqrt{  2\pi }}y^{1/2-1}e^{-y/2},\ y>0.

Since \Gamma \left( 1/2\right) =\sqrt{\pi }, the procedure for identifying the gamma distribution gives

f_{\chi _{1}^{2}}\left( x\right) =\frac{1}{\Gamma \left( 1/2\right) }\left(  1/2\right) ^{1/2}x^{1/2-1}e^{-x/2}=f_{1/2,1/2}\left( x\right) .

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 \chi _{n}^{2}=z_{1}^{2}+...+z_{n}^{2} with independent standard normals z_{1},...,z_{n} we can write \chi _{n}^{2}=\chi _{1}^{2}+...+\chi _{1}^{2} 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.

Recall that the gamma density is closed under convolutions with the same \alpha . Then by the convolution theorem we get

f_{\chi _{n}^{2}}=f_{\chi _{1}^{2}}\ast ...\ast f_{\chi  _{1}^{2}}=f_{1/2,1/2}\ast ...\ast f_{1/2,1/2} =f_{1/2,n/2}=\frac{1}{\Gamma \left( n/2\right) 2^{n/2}}x^{n/2-1}e^{-x/2}.