types of log two–piece distributions have not been studied to the best of our knowledge. The corresponding CDF is given by $$\begin{aligned}
\label{LTPCDF}
S_l(y; \mu,\sigma,\gamma,\delta) &=& \dfrac{2b(\gamma)}{a(\gamma)+b(\gamma)}  S\left(\dfrac{\log(y)-\mu}{\sigma b(\gamma)};\delta\right)I(y<e^{\mu}) \notag\\
&+&\left[\dfrac{b(\gamma)-a(\gamma)}{a(\gamma)+b(\gamma)} + \dfrac{2a(\gamma)}{a(\gamma)+b(\gamma)}S\left(\dfrac{\log(y)-\mu}{\sigma a(\gamma)};\delta\right)\right]I(y\geq e^{\mu}) .\end{aligned}$$

We can observe that the ratio of the mass cumulated on either side of the value $y=e^{\mu}$ is given by $$\begin{aligned}
R(\gamma) &=& \dfrac{S_l(e^{\mu}; \mu,\sigma,\gamma,\delta)}{1-S_l(e^{\mu}; \mu,\sigma,\gamma,\delta)} = \dfrac{b(\gamma)}{a(\gamma)}.\end{aligned}$$

This helps us to identify the different roles of the parameters $\gamma$ and $\delta$. The parameter $\gamma$ controls the cumulation of mass on either side of $y=e^{\mu}$, while the parameter $\delta$ controls the tails of the density. In Figure \[fig:LTPN\]a we present some examples of a two-piece normal PDF with different values of the parameter $\gamma$. In these cases, the parameter $\gamma$ only affects the asymmetry of the density. Figure \[fig:LTPN\]b shows the corresponding LTP normal PDFs. We can observe that in these cases the parameter $\gamma$ affects the shapes of the density. That is, it controls the mass cumulated above and below the value $y=1$ as well as the spread and mode of the density. The corresponding hazard function can be easily constructed from (\[LTPPDF\]) and (\[LTPCDF\]). Figure \[fig:DH\] shows the variety of shapes of 