Derivation of snedecors F distribution

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SUMMARY

The F distribution is derived from two independent chi-squared distributions, specifically when \(X_1 \sim \chi_{n_1}^2\) and \(X_2 \sim \chi_{n_2}^2\). The relationship is defined as \(\frac{X_1/n_1}{X_2/n_2} \sim F(n_1,n_2)\). This derivation includes the moment-generating function (MGF) and the calculation of raw moments, mean, and variance of the F distribution. Understanding this theorem is crucial for statistical analysis involving variance ratios.

PREREQUISITES
  • Understanding of chi-squared distributions, specifically \(\chi_{n_1}^2\) and \(\chi_{n_2}^2\).
  • Familiarity with the concept of independent random variables.
  • Knowledge of moment-generating functions (MGF).
  • Basic statistics, including mean and variance calculations.
NEXT STEPS
  • Study the properties of the F distribution in statistical inference.
  • Learn about the derivation and applications of moment-generating functions (MGF).
  • Explore the relationship between chi-squared distributions and the F distribution in depth.
  • Investigate statistical tests that utilize the F distribution, such as ANOVA.
USEFUL FOR

Statisticians, data analysts, and researchers involved in hypothesis testing and variance analysis will benefit from this discussion.

coolamko12
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derive f distribution as x1 and x2 are independent random variables having a chi-square distribution with n1 and n2 degrees of freedom.. aslo mgf and find raw moments ... and finds its mean and variance.?prove it ?help me guys?
 
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coolamko12 said:
derive f distribution as x1 and x2 are independent random variables having a chi-square distribution with n1 and n2 degrees of freedom.. aslo mgf and find raw moments ... and finds its mean and variance.?prove it ?help me guys?

There's a theorem which states that the $F$ distribution can be defined as the fraction of two independent chi-squared distributions. More precisely,

If $X_1 \sim \chi_{n_1}^2$ and $X_2 \sim \chi_{n_2}^2$ are two independent chi-squared distributions then $\frac{X_1/n_1}{X_2/n_2} \sim F(n_1,n_2)$.
 

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