I thought I understood the Memoryless property for Exponential Distributions but I am unable to get the intuition behind the answer to this question:
You are waiting for a bus at a bus station. The buses arrive at the station according to a Poisson process with an average arrival time of 10 mins. If the buses have been running for a long time and you arrive at the bus station at a random time, what is your expected waiting time. On average, how many minutes ago did the last bus leave?
Clearly expected waiting time is 10 mins. However, I dont get this part: "On average, how many minutes ago did the last bus leave". The explanation in the book is that if you look back in time, the memoryless property implies on average the last bus arrived 10 mins ago as well. I'm basically struggling to see how the memoryless property implies this above statement.
Thanks
Best Answer
I have to say at the start that bus arrivals do not typically follow an exponential distribution. So it is really hard to get out of your mind how $actual$ buses work, if someone says interarrival times are governed by an exponential process.
Maybe it is easier to think about something that really is exponentially distributed. Suppose you have a very weak radioactive source and you are capturing particles it emits in a counter. Suppose that the detection rate is one per 10 seconds. If you start keeping time at one particular click of the counter, then the average wait for the next click is 10 seconds.
However, the no-memory property says if we start keeping time at some arbitrary point in time (click or not), the average wait until the next click is also 10 seconds. The decaying particles are not 'keeping track' of each other, and they don't 'know' when you start counting.
Now suppose you have a paper tape on which clicks are recorded along a time line. The tape will look pretty much the same whether you read forwards or backwards: random marks spaced sometimes near together, sometimes relatively far apart, but $on\; average$ 10 seconds apart. Maybe it is possible to say this is due to the no-memory property, but in my experience the usual terminology for this is 'time-reversibility'.
Both no-memory and time-reversibility are fundamental properties of exponential processes, so I suppose it is possible to take a point of view (for exponential processes) that your statement in bold type is true. But I'm not sure there is a lot of intuitive value in trying to make this connection between memorylessness and time reversibility when you're just starting to think about the curious properties of exponential models.
As another example on no-memory, suppose a computer unit in a satellite survives an exponentially distributed length of time with mean lifetime 10 years. Radiation hits are what cause such computer units to die. If 8 years have already gone by, you might think the computer unit is nearing the end of its life. But if the lifetime really is exponentially distributed, and it is still alive at 8 years, the expected time of death from a random radiation hit is still 10 years away. For such devices, we say "Used is as good as new." This is an appropriate model for devices that die only because of random radiation hits. (Of course if you have a census of dead satellite computers of this type along with their 'death' dates, you could check back to their 'birth' dates and see that they were, on average, about 10 years before the death dates, but that is not much of a profound statement.)
For humans in a certain population we might say that their average lifetime at birth is 70 years. If such a person is now 60 years old, it would not be reasonable to say that he or she has another expected 70 years of life. People do sometimes die of random accidents, but they also die by 'wearing out' with age.
Most things we are familiar with die from a combination of random accidents and gradual wearing out: automobiles, light bulbs, pets, T-shirts, and so on. Other events, like elections, bus arrivals, credit card bills, and so on tend to happen at rather even intervals--sometimes without much of a random component. The reason intuition comes so hard when thinking about exponentially distributed events is that there are relatively few events in real life that happen according to an exponential model.
In science things get modeled according to exponential distributions for two reasons: (a) Some things really are exponentially distributed--at least approximately. Service times at banks, lives of transistors, radioactive decay, and so on. (b) Because the no-memory rule makes it unnecessary to take past history into account, exponential models are mathematically very easy to handle; that makes it tempting to use exponential models sometimes when they don't really apply very well.