Chi-Squared Distribution

Very few real-world observations follow a chi-square distribution. The main purpose of chi-square distributions is Hypothesis Testing, not describing real-world distributions.

Definition 1:
Let be a positive integer. Then we say if

  • , ’s are independent
  • a parameter that represents the Degrees of Freedom

Definition 2 using the PDF: where

Expectation and Variance

If we apply CLT, we get that as ,

  • table

Chi-Squared Table


So in here, unlike in the Z-Table, you are given a degree of freedom and a probability, and this table gives you the value.

Since is one-tailed, then you just check the 95%,

Ex1: df =15, percentile = 0.9, you find that x = 22.307 Ex2: Let , Find and such that

You can consider the probabilities and , so you have .