Probstat/notes/chi-squared distribution
- This is part of probstat.
Definition
Definition: Let be independent unit normal random variables. A random variable
is called a chi-squared random variable with degrees of freedom. We also write
Wikipedia has a nice article on chi-squared distribution which also includes plots of its pdf and cdf.
Properties
Here we states important properties of the chi-squared distribution without proofs.
Expectations and variances
If is chi-squared with degrees of freedom, we have that its expectation
![{\displaystyle E[X]=n,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0669d63dd3dfd0db0e9bfd7ee2814b00f7b74e48)
and its variance
Sample variances
The chi-squared distribution is very important when we want to reason about the sample variances (See notes). Under that settings, we sample from a normal population whose mean is and variance is .
We first note that if and is known, a term similar to the one we used to compute the same variance is chi-squared. Recall that is unit normal. Therefore,
However, usually, is unknown and we are left to the sample variance, defined as
.
Note that the only differences between this term and the previous one is the use of sample mean instead of the population mean. It can be proved (not here) that in fact, for normal population, (with some scaling) is actually a chi-squared random variable with degrees of freedom (instead of as in the term computed from ). That is,
More over, for normal population, and are independent. This is very surprising because both quantities are calculated from the same sample.
These two facts are important in the development of the procedure for calculating confidence intervals in case where is unknown. (See notes.)
Two important facts about sample means and sample variances:
1.
2. and are independent.
