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Mathematics

Probability

Chi-Squared Distribution

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Degrees of Freedom

The number of independent values that can vary in the calculation of a statistic; for Chi-Squared, it's the number of standard normal variates being summed.

Use Case: Test for Independence

Used in categorical data to assess whether variables are independent of one another in a contingency table.

Definition of Chi-Squared Distribution

A distribution that models the sum of the squares of k independent standard normal random variables.

Characteristic Function of Chi-Squared

Given by $\phi(t) = (1 - 2it)^{-k/2}$ , where t is a complex number and k is the degrees of freedom.

Probability Density Function (PDF)

Given by

f(x;k) = \frac{1}{2^{k/2}\Gamma(k/2)}x^{(k/2)-1}e^{-x/2}

for x > 0, where

\Gamma

is the gamma function and k is the degrees of freedom.

Use Case: Goodness-of-Fit Test

The Chi-Squared distribution is used to determine how well observed data fit a model's expectations.

Chi-Squared Distribution in Regression Analysis

Used to test for the significance of independent variables in a model, especially in the context of the ANOVA (Analysis of Variance) technique.

Moment Generating Function (MGF)

The MGF of a Chi-Squared distribution is given by $M(t) = (1 - 2t)^{-k/2}$ , provided that $t < \frac{1}{2}$ .

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