New stochastic inequalities involving the F and Gamma Distributions
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We derive a new inequality involving either the F or Gamma distribution and its quantiles and call it either the F-Inequality or G-Inequality. The stochastic representation of the new inequality involves α, p ∈ (0, 1) such that if p > α and k > 1 with W being a random variable with an F(v<sub>1</sub>, v<sub>2</sub>) or Gamma(τ, θ) distribution, then 1/pP(W < W<sub>p</sub>/k) > 1/αP(W < Wα/k), where for any γ between 0 and 1, W<sub>γ</sub> is defined by α = P(W < W<sub>α</sub>). The inequality reverses for k ∈ (0, 1); it becomes equality for k = 1 and, trivially, for k = ∞. This inequality seems to hold for a larger class of distributions that include for example the F, Gamma, Cauchy, and special cases of the Beta distribution and perhaps others. However, we provide rigorous proofs only for the F and Gamma distributions.