I've read that time between calls (in a call center) can be modeled with exponential distribution. My question is this: the shape of the exponential distribution has a decreasing nature. Suppose that there are 6 calls at average in an hour. In this case, according to my intuition, the highest probability for the next call to happen (after a call) should be 10 minutes. Any time smaller than 10 minutes should have a lower probability compared to 10 minutes. So, the shape of the distribution should not be decreasing but should be something like achieving its highest value around 10. I would be glad if you point out what is wrong in my reasoning.
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I do not see that as intuitive. Suppose the calls were equally spaced - then they would be $10$ minutes apart. But in this case they are not equally spaced (no caller knows what times the other callers choose) so the gaps could be much larger or could be shorter. There is a lower bound of $0$ on how short a gap can be but no upper bound on how long, so to keep the average gap at $10$ minutes and allow for possible very large gaps, the distribution would be skewed and the most likely gap should be less than $10$ minutes (and is in fact arbitrarily close to $0$) – Henry Sep 03 '22 at 15:57
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You appear to conflate the expectation and the mode. – whuber Sep 04 '22 at 15:31
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This is very helpful @Henry, thank you. – Sanyo Mn Sep 04 '22 at 17:05