$($a$)$ Describe in words the following present value random variables of

an insurance policy issued to $(50)$ :

$Y_1=\{\begin{array}{cc}{a}_{\frac{?}{\text{b}\text{a}\text{r}}(201)|},& K_{50}20\end{array}$

$a_{\frac{?}{\text{b}\text{a}\text{r}}(K_{50})},K_{50}>20$

$Y_2=\{\begin{array}{cc}{}_{\frac{?}{\text{b}\text{a}\text{r}}(T_{50}|)},& T_{50}20\end{array}$

${}_{\frac{?}{\text{b}\text{a}\text{r}}(201)},T_{50}>20.$

$($b$)$ Write their expected present values in terms of actuarial functions.

$($c$)$ Calculate their expected present values if $=0\mathrm{.}5$ and ${}_{x+t}=0\mathrm{.}3$ for all

ages $x$ and all $t0.$