## 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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