An individual has a transition intensity of mortality for the next 10 years of μs=0.01s per annum (where s is the number of years from today). Calculate the probability (to 3 decimal places) that they will survive for the next 10 years
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To calculate the probability of survival for the next 10 years, we can use the concept of survival probability and the transition intensity of mortality.
The transition intensity of mortality (μs) is the rate at which an individual is expected to transition from being alive to dying, per annum. In this case, the transition intensity is μs = 0.01s per annum, where s is the number of years from today.
The survival probability (Ps) at a given time is equal to 1 minus the cumulative probability of dying up to that time. Mathematically, it can be written as:
Ps = exp(-∫μs ds)
Here, the integration is with respect to time (s) from 0 to 10 years.
To solve this equation, we need to calculate the integral of μs with respect to s over the range of 0 to 10 years.
∫μs ds = ∫0.01s ds = 0.01 * 0.5 * s^2 (evaluated from 0 to 10)
∫μs ds = 0.01 * 0.5 * (10^2 - 0^2)
= 0.01 * 0.5 * 100
= 0.01 * 50
= 0.5
Using this value, we can now calculate the survival probability (Ps) for the next 10 years:
Ps = exp(-∫μs ds)
= exp(-0.5)
Using a calculator, we find that exp(-0.5) is approximately 0.606.
Therefore, the probability of surviving for the next 10 years is approximately 0.606 to 3 decimal places.