excans:77fb6df52e

Answer: A

[math]E\left[T_{0}\right]=\int_{0}^{8}{ }_{t} p_{0} d t=\int_{0}^{8}\left(1-\frac{t^{2}+t}{72}\right) d t \rightarrow \frac{1}{72}\left[72 t-\frac{t^{3}}{3}-\frac{t^{2}}{2}\right]_{0}^{8}=5.1852[/math]

[math]E\left[T_{0}{ }^{2}\right]=2 \int_{0}^{8}\left({ }_{t} p_{0} \times t\right) d t=\frac{2}{72} \int_{0}^{8}\left(72 t-t^{3}-t^{2}\right) d t=\frac{2}{72}\left[36 t^{2}-\frac{t^{4}}{4}-\frac{t^{3}}{3}\right]_{0}^{8}=30.815[/math]

[math]\operatorname{Var}\left[T_{0}\right]=E\left[T_{0}^{2}\right]-\left(E\left[T_{0}\right]\right)^{2}=3.9287[/math]

Copyright 2024 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.