Fig. 1.

Different BS PDFs according to different parametrization. In red $\alpha =0.5\text{and}\beta =1$; in green $\alpha =0.8\text{and}\beta =1$; in blue $\alpha =1.5\text{and}\beta =1$; and in cyan $\alpha =0.2\text{and}\beta =1.5$. We can notice that in the last case, $\alpha$ is small, and then its shape is more symmetrical than the former cases. This is because its expectation, $\beta (1+{\alpha }^2/2)$ is near its median $\beta$. (Color figure online)