Last modified: September 21, 2024
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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.