Chi-Square Distribution (Continuous)

Last modified: February 26, 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 $X \sim χ^{2} (k)$ , where k is the number of degrees of freedom.

Probability Density Function (PDF)

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

$f (x) = {\begin{cases} \frac{1}{2^{k / 2} Γ (\frac{k}{2})} x^{\frac{k}{2} - 1} e^{- \frac{x}{2}}, & if x \geq 0 \\ 0, & if x < 0 \end{cases}$

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

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Cumulative Distribution Function (CDF)

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

$F (x) = \frac{γ (\frac{k}{2}, \frac{x}{2})}{Γ (\frac{k}{2})}$

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

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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:

$Var (X) = 2 k$

Moment Generating Functions and Moments

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

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

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:

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

First Moment (Mean):

$E [X] = k$

Second Moment (Variance + Mean^2):

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

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:

$χ^{2} = \sum \frac{(O_{i} - E)^{2}}{E}$

where $O_{i}$ is the observed frequency and $E$ is the expected frequency.

I. Calculate the chi-square statistic:

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

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.