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Consider someone waiting for a bus. We know how the time ($T$) that a new arriver at the bus stop waits is distributed based on historical data (let's say they are normally distributed with mean 15 and standard deviation 2). So we know that the expected waiting time is 15 minutes.

Given that someone has been waiting ($W$) 14 minutes, can we say that the expected remaining waiting time is 1 minute?

Or is the probability different since we now want the wait time conditional on having already waited 14 minutes? i.e. We now want $P(T=15|W=14)$.

Adam Kells
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    https://stats.stackexchange.com/questions/166273/expected-value-of-x-in-a-normal-distribution-given-that-it-is-below-a-certain-v – ocram Feb 02 '22 at 11:11
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    The problem description is unclear. You say that expected waiting time is 15m, so you assume that the previous bus has just left? But than, you say that the person has already been waiting for 14m, since what? Since the person came to the bus stop, or since the previous bus left? When did the person come to the bus stop? – Tim Feb 02 '22 at 11:26
  • I've edited to clarify that we are not interested in the time between buses and only have data on previous waiting times for people who arrived at the bus stop. – Adam Kells Feb 02 '22 at 13:56
  • You don't have the information needed to deduce conditional probabilities. It is instructive to study the simplest case of *independent,* *exponentially distributed* waiting times (a *homogeneous Poisson process*), even if you believe they don't apply to your situation, because it reveals the importance of subtle properties such as stationarity, ergodicity, and independence of events. – whuber Feb 02 '22 at 15:43

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