The following construction formalizes a useful symmetry: two scaled Beta distributions can be designed to have identical supports, means, and medians, yet opposite skewness. Such distributions represent mirror-complementary distortions of a common latent shape and serve as a controlled test bed for quantifying divergence in the presence of asymmetric deformation.
Let \(Z_A \sim \mathrm{Beta}(\alpha,\beta)\) and \(Z_B \sim \mathrm{Beta}(\beta,\alpha)\) with \(\alpha,\beta>0,\ \alpha\neq\beta\). Define a symmetric interval \([x_{\min},x_{\max}]\) with midpoint \(x_c=\tfrac{x_{\min}+x_{\max}}{2}\) and scale \(s=x_{\max}-x_c\). Set
\[ X_A = x_c + s\,(Z_A - \tfrac{1}{2}), \qquad X_B = x_c - s\,(Z_A - \tfrac{1}{2}). \]
Both \(X_A\) and \(X_B\) lie in \([x_{\min},x_{\max}]\); one has a heavy left tail, the other a heavy right tail, while their means and medians coincide at \(x_c\).
Let \(X_A, X_B\) be defined as above. Then \[ \mathbb{E}[X_A]=\mathbb{E}[X_B]=x_c, \qquad \mathrm{median}(X_A)=\mathrm{median}(X_B)=x_c, \] and their standardized skewness coefficients satisfy \[ \gamma_1(X_B) = -\,\gamma_1(X_A) = -\,\frac{2(\beta-\alpha)\sqrt{\alpha+\beta+1}} {(\alpha+\beta+2)\sqrt{\alpha\beta}}. \] Thus, \(X_A\) and \(X_B\) possess identical first-order location statistics but opposite asymmetry about \(x_c\).
Intuition. The transformation simply reflects a Beta density about its midpoint, preserving all even moments and inverting all odd ones. This isolates directional asymmetry as the only distinguishing factor.
Let \(f_A\) and \(f_B\) be the densities of \(X_A\) and \(X_B\). Under the reflection condition \(f_B(x)=f_A(2x_c-x)\), \[ \mathrm{TV}(f_A,f_B) = \tfrac{1}{2}\!\int_{x_{\min}}^{x_{\max}}\! |f_A(x)-f_B(x)|\,dx \] \[ = 1 - 2\!\int_{x_c}^{x_{\max}}\! \min\{f_A(x),f_A(2x_c-x)\}\,dx. \] This yields the minimum upper bound
\[ \mathrm{TV}_{\min}=1 - 2\!\int_{x_c}^{x_{\max}}\! f_A^{\mathrm{sym}}(x)\,dx, \] \[ f_A^{\mathrm{sym}}(x)=\min\{f_A(x),\,f_A(2x_c-x)\}. \]
Equality (\(\mathrm{TV}=0\)) occurs only for perfectly symmetric shapes \((\alpha=\beta)\); increasing asymmetry \(|\alpha-\beta|\) enlarges the distance.
Intuition. Total variation depends solely on how much probability mass fails to overlap under reflection. When the distributions are mirror images, overlap is maximal and the difference purely directional.
For small deviations \(\delta=\tfrac{\beta-\alpha}{\alpha+\beta}\) around symmetry (\(\alpha\approx\beta\)), the total-variation distance between opposite-skew Beta pairs satisfies
\(\displaystyle \mathrm{TV}(f_A,f_B) \approx \frac{4}{3\pi}\,|\delta|\, \sqrt{\tfrac{\alpha+\beta+1}{\alpha+\beta+2}} \propto|\alpha-\beta|. \)
Hence, the distance grows linearly with parameter imbalance for near-symmetric Betas, and quadratically in the corresponding skewness magnitude \(|\gamma_1|\).
Intuition. At small asymmetry, the reflected pair differs only in the sign of an odd perturbation term in the log-density expansion, so total variation scales linearly with \(|\alpha-\beta|\).
Together, these results offer a compact formalization of mirror-asymmetric equivalence: two Beta-shaped distributions can share identical mean, median, and support, yet exhibit opposite skewness in form. This reflection symmetry yields a clean analytic handle on asymmetry-induced divergence. The total-variation bound quantifies the irreducible distance due solely to tail reversal, while the corollary connects that distance directly to the Beta parameters, providing a controllable, analytic test case for asymmetry sensitivity in statistical models.