Write down the contribution to the likelihood from to for each of the life histories described in the following table, each of whom enters observation at age x. Time t is counted from age x.

Two Math Questions
Problem 2.
The following data are taken from the employment records between ages 58 and 59 of 1000 pension plan members. The possible decrements from active employment (State 0) are by early retirement (State 1), transfer to another position (State 2) or death in service (State 3).
You are give.
Total waiting time in active employment: 785 years
Total number of early retirements: 150
Total number of withdrawals: 200
Total number of deaths in service: 20
Problem 3.
A long-term care insurance policy is modelled by the following four-state model with constant transition inten.,:ties between integer ages: Able (0), Partially able (1), Disabled (2), Dead (3). Possible transitions are: 0 —> 1, 1 —> 2, 0 3, 1 —> 3, and 2 —> 3.
a). Write down the contribution to the likelihood from to for each of the life histories described in the following table, each of whom enters observation at age x. Time t is counted from age x.
Life 1 I Life 2
Life 3
t = 0 In State 1 t = 1 In State 1
t = 0 In State 0 t = 0.25 Moves to State 1 t = 0.75 Moves to State 2
t = 0 In State 1 t = 0.5 Moves to State 2 t = 0.8 Dies
b). The actuary has the following summary of all observations: — Total waiting time in State 0: 4521.2 years — Total waiting time in State 1: 357.9 years — Total waiting time in State 2: 69.6 years — Total number of transitions 0 -4 1: 163 — Total number of transitions 0 3: 125 — Total number of transitions 1 2: 154 — Total number of transitions 1 3: 42 — Total number of transitions 2 -4 3: 25 Derive maximum likelihood estimates of all the transition intensities along with estimates of the as-sociated standard errors.

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