A cable car starts off with n riders. The times between...

A cable car starts off with $n$ riders. The times between successive stops of the car are independent exponential random variables with rate $\lambda$. At each stop one rider gets off. This takes no time, and no additional riders get on. After a rider gets off the car, he or she walks home. Independently of all else, the walk takes an exponential time with rate $\mu$. (a) What is the distribution of the time at which the last rider departs the car? (b) Suppose the last rider departs the car at time $t$. What is the probability that all the other riders are home at that time?

A cable car starts off with $n$ riders. The times between successive stops of the car are independent exponential random variables with rate $\lambda$. At each stop one rider gets off. This takes no time, and no additional riders get on. After a rider gets off the car, he or she walks home. Independently of all else, the walk takes an exponential time with rate $\mu$.
(a) What is the distribution of the time at which the last rider departs the car?
(b) Suppose the last rider departs the car at time $t$. What is the probability that all the other riders are home at that time?

Key Concepts

Exponential Distribution

The exponential distribution is used to describe the time between independent events that occur continuously and independently at a constant average rate. Its key characteristic is the memoryless property, which means that the probability of an event occurring in the future is independent of how much time has already elapsed. This property greatly simplifies the modeling of waiting times in various stochastic processes.

Gamma Distribution

The gamma distribution arises as the sum of independent exponential random variables. When you add a number of independent exponential waiting times, the resulting total waiting time follows a gamma distribution, with its shape parameter corresponding to the number of summed variables and its rate parameter matching that of the individual exponentials. This concept is central to understanding cumulative event times in processes with sequential steps.

Independence of Random Variables

Independence between random variables means that the occurrence or outcome of one event does not impact the probability distribution of another. In many stochastic models, such as those involving waiting times or service times, assuming independence allows each stage or component of the process to be analyzed separately, simplifying the overall system analysis and probability computations.

Conditional Probability in Stochastic Processes

Conditional probability in stochastic processes is the likelihood of an event occurring given that another event is known to have occurred. This approach is crucial when analyzing processes where certain outcomes are revealed or fixed, as it helps update the probabilities of related events. In models with exponential waiting times, the memoryless property further facilitates conditional calculations, as the process effectively 'restarts' at the conditioning point.

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