Final answer:
The time it takes an individual to retire after age 60 follows an exponential distribution with a mean of five years, denoted as X~Exponential(1/5). To find specific probabilities or expectations, the probability density and cumulative distribution functions for the exponential distribution are utilized.
Step-by-step explanation:
The time after reaching age 60 that it takes for an individual to retire is given as an exponential distribution with a mean of five years. Since the question describes the time until an event (retirement) occurs, and it has a known average time (mean), selecting the exponential distribution option is appropriate. Thus, we can denote this as X being a random variable representing retirement time, following an exponential distribution with a rate parameter (λ) that is the inverse of the mean, so λ = 1/5.
The correct choice for this scenario would be (a) Exponential distribution with mean 5. Probability density function of an exponential distribution is described as f(x; λ) = λ * e^{-λ x} for x ≥ 0. To calculate the probability that a person retired after 70, we consider the time to retirement to be 10 years after 60 (70-60), and use the exponential distribution's cumulative distribution function to find the probability that X > 10. More people are likely to retire before 65 due to the decreasing nature of the exponential probability curve.
In a room of 1,000 people over age 80, the expectation would be very high that nearly all would have retired, given a mean retirement age of 65. However, to find a specific number, one would apply the exponential distribution probability formula to calculate the chance that someone has not retired by 80, then multiply that probability by 1,000.