chi_square_distribution.md

Chi-Square Distribution (Continuous)

A chi-square distribution is a continuous probability distribution of the sum of the squares of k independent standard normal random variables. The chi-square distribution is denoted as $X \sim \chi^2(k)$, where k is the number of degrees of freedom.

Probability Density Function (PDF)

The PDF of a chi-square distribution is given by:

$$f(x) = \begin{cases} \frac{1}{2^{k/2} \Gamma \left(\frac{k}{2}\right)} x^{\frac{k}{2} - 1} e^{-\frac{x}{2}}, & \text{if}\ x \ge 0 \\ 0, & \text{if}\ x < 0 \end{cases} $$

where $\Gamma(\cdot)$ is the gamma function.

Cumulative Distribution Function (CDF)

The CDF of a chi-square distribution is given by:

$$F(x) = \frac{\gamma \left(\frac{k}{2}, \frac{x}{2}\right)}{\Gamma\left(\frac{k}{2}\right)}$$

where $\gamma(\cdot)$ is the lower incomplete gamma function and $\Gamma(\cdot)$ is the gamma function.

Expected Value and Variance

The expected value (mean) of a chi-square distribution is equal to the number of degrees of freedom:

$$E[X] = k$$

The variance of a chi-square distribution is given by:

$$\text{Var}(X) = 2k$$

Moment Generating Functions and Moments

The moment generating function (MGF) of a chi-square distribution is:

$$M_X(t) = E[e^{tX}] = (1-2t)^{-\frac{k}{2}}$$

To find the n-th moment, we take the n-th derivative of the MGF with respect to t and then evaluate it at t=0:

$$E[X^n] = \frac{d^n M_X(t)}{dt^n}\Bigg|_{t=0}$$

• First Moment (Mean):

$$E[X] = k$$

• Second Moment (Variance + Mean^2):

$$E[X^2] = k^2 + 2k$$

Example: Goodness-of-Fit Test for a Fair Die

A researcher is testing the fairness of a six-sided die. The die is rolled 60 times, resulting in the following observed frequencies for each face:

Face of the Die	Observed Frequency
1	8
2	9
3	13
4	7
5	12
6	11

To determine if the die is fair, the researcher employs a chi-square goodness-of-fit test.

Given:

The expected frequency for each face (assuming the die is fair): E = Total rolls / Number of faces = 60 / 6 = 10

The chi-square formula:

$$ \chi^2 = \sum \frac{(O_i - E)^2}{E} $$

where $O_i$ is the observed frequency and $E$ is the expected frequency.

I. Calculate the chi-square statistic:

$$ \chi^2 = \frac{(8-10)^2}{10} + \frac{(9-10)^2}{10} + \frac{(13-10)^2}{10} + \frac{(7-10)^2}{10} + \frac{(12-10)^2}{10} + \frac{(11-10)^2}{10} = 3.6 $$

II. Determine the degrees of freedom:

Degrees of freedom for chi-square test: Number of categories - 1 = 6 - 1 = 5

Applications

Chi-square distributions are widely used in hypothesis testing (such as the chi-square test for independence), goodness-of-fit tests, and confidence interval estimation for the population variance in normally distributed data.