## 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. Assuming that , on Chart 1 we see that Hence, using additivity of probability,

(1)

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

(2)

**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.

### General chi-squared

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.

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

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