I'm studying actuarial mathmatics. Can you help me solving this question?
The PV random variable of $u$-year deferred whole life insurance benefit is
$Z = \begin{cases} 0, & \mbox{$T_x$ < $u$} \\ v^{T_x}, & \mbox{$T_x$ $\ge$ $u$} \end{cases}$
Note that discount factor $v=e^{-\delta}$. and $Z$ is decreasing function of $T_x$ when $T_x >u$. Assuming constant force of mortality $\mu$ and force of interest $\delta$, How can I derive the Cumulative distribution function(CDF) of $Z$ and calculate median of $Z$?
This is my idea: let $T_x = t$.
$f_x(t) = \mu e^{-\mu t},\, F_x(t) = {_t}q_x = 1-e^{-\mu t},\, S_x(t) = {_t}p_x = e^{-\mu t}$.
$P(Z=0) = P(T_x \leq u) = 1-e^{-\mu u}$ $P(Z=v^t) = {_u|_t}q_x = {_u}p_x{_t}q_{x+u} = e^{-\mu t}(1-e^{-\mu t}) = e^{-\mu t} - e^{-\mu (u+t)}$
but I cannot express it for function of $Z$. I got probabilities for $Z$, but they are also function of $t (T_x)$. CDF should be $f: Z \rightarrow [0,1]$, but there is another variable $t$. Could you help me how can I get appropriate function for $Z$, please?
Thank you for your help.