1. (14 points) Let X denote the time to an event of interest, the notations are as introduced in class. (a) (2 points) Please show that life expectancy E(X) = J. S(x)dx. (b) (2 points) Please show that mean truncated /restricted lifetime E(min(X, T)) = Jo S(x)dr. (c) (2 points) Please show that mean residual lifetime E(X - ro|X > To) = J. S(x)dx/ S(xo) (d) (8 points) Suppose that the time to death (in months) for a type of genetically engineered mouse exposed to a high dose of radiation follows an exponential distribution with constant hazard rate A = 2 per month. In this cohort of mice, find (a) (2 points) the expectation of life for a mouse, (b) (2 points) the (restricted) average survival time to two months, where two months is a prespecified time point when study may end and all mice will be sacrificed (c) (2 points) the average lifetime beyond two months for a mouse alive at two months, and (d) (2 points) the median time to death