Mohie El-Din and Amein (2011) define a distribution in formula (1.2)
which they call the exponential Bernoulli distribution (EBD).
The distribution has the following form:
$$\displaystyle f \left(t \right) = \left(1-p \right)~\alpha ~e^{-\alpha ~t }+p~\left(\alpha +\beta \right)~e^{-\left(\alpha +\beta \right)~t }$$
or with $\lambda = \alpha+\beta$ the distribution can also be written as :
$$\displaystyle f \left(t \right) = \left(1-p \right)~\alpha ~e^{-\alpha ~t }+p~\lambda ~e^{-\lambda ~t }$$
In Mohie El-Din and Amein (2011) these densities are written in $x$.
The logic behind this distribution is as follows. The Bernoulli variable $X$
with $X = 0, 1$ has a probability distribution with $P (X = 1) = p$
and $P (X = 0) = (1-p)$. Furthermore, there is a random variable $T$
with $0 \leq T$. If $X = 1$, then $T $ has an exponential distribution with rate parameter $\lambda$.  If $X = 0$, then $T $ has an exponential distribution with rate parameter $\alpha$.

Now I have another case. In my case the following applies. If  $X = 1$
then $T$ also has an exponential distribution:
$$\displaystyle f \left(t \right) = p ~\lambda ~e^{-\lambda ~t }$$
But if $X = 0$, then $T = 0$. The final distribution is a proper probability
distribution because:
$$\displaystyle 1-p+\int_{0}^{\infty }\!p \, \lambda\,{{\rm e}^{-\lambda\,t}}\,{\rm d}t=1-p+p\int_{0}^{\infty }\!\lambda\,{{\rm e}^{-\lambda\,t}}\,{\rm d}t \, = \, 1$$
The expectation of $T$ can be written as:
$$\displaystyle \mu_{{1}}\,  =  0 \,  (1-p) \, + \, \int_{0}^{\infty }\!p \, t \, \lambda\,{{\rm e}^{-\lambda\,t}}\,{\rm d}t  \,= \, \frac {p}{\lambda}$$
The variance of $T$ can be written as:
$$\displaystyle \mu_{{2}}\,  =  \, (1-p) \,  \left(0-\frac{p}{\lambda}\right) ^2 +\int_{0}^{\infty }\!p \, \left( t-{\frac {p}{\lambda}} \right) ^{2}\lambda\,{{\rm e}^{-\lambda\,t}}\,{\rm d}t \,  =  \,-{\frac {p \left( p-2 \right) }{{\lambda}^{2}}}$$
The third central moment can be written as:
$$\displaystyle \mu_{{3}}\,  =  \, (1-p) \,  \left(0-\frac{p}{\lambda}\right) ^3 +\int_{0}^{\infty }\!p \, \left( t-{\frac {p}{\lambda}} \right) ^{3}\lambda\,{{\rm e}^{-\lambda\,t}}\,{\rm d}t \,  =  \,2\,{\frac {p \left( {p}^{2}-3\,p+3 \right) }{{\lambda}^{3}}}$$
I now have three questions. The first question is: what is the best way
to name this distribution? The second question is: how can I best write
down this distribution in a formula? The third question is: are the
derivations of the three given central moments correct?

Reference

Mohie El-Din, M. M. and Amein, M. M. (2011). Estimation of Parameters of the Exponential Bernoulli Distribution Based on Progressively Censored Data. _Applied Mathematical Sciences_, Vol. **5**, no. **58**, 2883 - 2890.