Figure 1

Dichotomy of the beta-exponential distribution relative to the value \(a=1\). Left: illustration of Corollary 1 displaying curves \(b\mapsto f_{\mathrm{BE}}(n,a,b)\), with a taking values 0.1 (blue curves), 1 (black curve), and 10 (red curves), and n taking values 1 (continuous curves), 2 (dashed curves), and 3 (dotted curves). Right: illustration of Proposition 1 beta-exponential density function \(g(x;a,1)\) for values of \(a\in [0.4,4]\): densities are log-convex for \(0< a<1\) (in blue), log-concave for \(a>1\) (in red), while \(a=1\) corresponds to the exponential distribution with mean 1 (in black)