※ 引述《scitamehtam (scitamehtam)》之铭言： : 在右偏的机率分配下 : 平均数 > 中位数 : 直观理解是因为极端值 : 对平均数的影响也比较大吗？ : 还是有其他直观的解释呢？ : 有证明吗？ : 谢谢 : ---- : Sent from BePTT on my iPhone 12 https://reurl.cc/pLyO2Q https://www.tandfonline.com/doi/full/10.1080/10691898.2005.11910556 这结果很多情况都不是对的... 尤其是discrete distribution: All Poisson distributions have an infinite right tail and positive skew (equal to ) – yet for more than 30% of parameter values, for example μ = .75 (Figure 5), the mean is less than the median. 即使是continous distribution: To construct a multimodal violation, simply take a discrete violation (e.g., Figure 2 or Figure 5) and add random normal “noise” to each value of X. The noise makes the distribution continuous, but if the noise variance is small there will be little change to the mean, median, mode, or skew 不过... Unimodal continuous densities are more cooperative. CitationGroeneveld and Meeden (1977) prove that the skew gives the relative positions of mean, median and mode for the F, beta, and gamma densities (the gamma includes the exponential and the chi-square). More generally, CitationMacGillivray (1981) proves the relationship for a large class of continuous unimodal densities including the entire Pearson family. eg: X~ exp(a) ∞ -x/a ∞ -u -u ∞ ∞ -u u u ∞ μ = ∫xe /a dx = a∫ ue du = a(-ue | +∫ e du) = a(-lim ---- -e |) 0 0 0 0 u->∞ e^u 0 1 = a(-lim ----- - (-1)) = a u->∞ e^u 2 ∞ 2 -x/a 2 ∞ 2 -u 2 2 -u ∞ ∞ -u E(X ) = ∫ x e /a dx = a ∫ u e du = a (-u e | + 2∫ue du) 0 0 0 0 2 u^2 2E(X) = a (-lim ---- + ------- ) u->∞ e^u a 2 2 = a (-lim ----- + 2) u->∞ e^u 2 = 2a 2 2 2 2 2 2 => σ = E(X )-μ = 2a - a = a X-μ 3 1 3 2 2 3 skew = E[(------) ] = ------E(X -3X μ+3Xμ -μ ) σ σ^3 1 3 2 2 3 = ------(E(X )-3μE(X ) +3μ E(X)-μ ) σ^3 1 3 3 3 3 = -----(E(X )-6a +3a -a ) a^3 1 3 3 = -----(E(X )-4a ) a^3 3 ∞ 3 -x/a 3 ∞ 3 -u 3 3 -u ∞ ∞ 2 -u E(X ) = ∫ x e /a dx = a ∫ u e du = a (-u e | + 3∫ u e ) 0 0 0 0 3 u^3 3E(X^2) = a (-lim ----- + --------) u->∞ e^u a^2 3 6 = a (-lim ------ + 6) u->∞ e^u 3 = 6a => skew = 2 > 0 m -x/a ∫ e /a dx = 1/2 0 -x/a m => -e | = 1/2 0 -m/a => 1-e = 1/2 => -m/a = ln(1/2) => m = aln2 < alne = a = μ 这个情况下的确skew > 0, μ > m --

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