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3.8.1 F Distribution

3.8.1 F - Distribution

Key Concepts

  • F-Distribution: A ratio of two scaled chi-square distributions which are independent.
  • Degrees of Freedom: Characterized by two parameters, k1 and k2, which are the degrees of freedom of the two chi-square distributions in the numerator and denominator, respectively.

Detailed Explanation

Formation of the F-Distribution

  • If U and V are two independent chi-square random variables with k1 and k2 degrees of freedom respectively, then the random variable
    • F = (U/k1) / (V/k2) follows an F-distribution with k1 and k2 degrees of freedom.

Properties of F-Distribution

  • Skewness: The distribution is not symmetric but skewed to the right, with the extent of skewness decreasing as k2 increases.
  • Mean and Variance: The mean of the F-distribution is k2 / (k2 - 2) for k2 > 2, and its variance is more complex, generally increasing with k1 and decreasing as k2 increases.

Applications

  • ANOVA: Used to compare the variances of three or more samples to find out if at least one sample mean differs significantly from the others.
  • Regression Analysis: Assists in determining whether variables are significant predictors of an outcome.

Practical Application

Example Calculation:

  • If you are testing the equality of variances across two independent samples where one sample has 10 degrees of freedom and the other has 20, you would use an F-distribution with k1 = 10 and k2 = 20.
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