What is the expected period of a recurring event that has a random (but limited) interval between each occurrence?
So we have this event. Amazing, right?
We also have a fixed maximum duration timer like one of these rotary kitchen timers. Not really great, but definitely comfortable.
When the timer goes off, an event occurs and the timer is reset to some random value between zero and the maximum duration (each value has an equal probability or is chosen). The timer starts immediately and the process continues forever.
My question is, what is the effective event period? That is, if another recurring event has a fixed period, so that this period should have an equal number of occurrences for two events over a very long period of time.
I mean, this is half the maximum timer duration, but I don't know why, and I wouldn't be surprised if it isn't.
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I mean, this is half the maximum timer duration, but I don't know why, and I wouldn't be surprised if it isn't.
Your intuition is correct. This is just a uniform selection of random numbers from the interval [0, N], where N is the maximum number of minutes on the timer. The expected value of this selection is the average of the minimum (0) and maximum (N) values, so the result is N / 2.
You can imagine some other similar cases; for example, the expected die cast value is 3.5 because the minimum value is 1 and the maximum value is 6, which gives (1 + 6) / 2 = 3.5.
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I think you are correct: 0.5 * maximum timer duration.
Reasoning: the maximum period with a given setting will be the maximum duration of the timer.
The average selected duration will be equal to the maximum timer period, if they all have equal probability: add them, divide by count and see for yourself:
Example: values 1, 2, 3, 4, 5, 6
Since each of them has the same probability of being selected, since N is large: 1 will be selected N / 6 times, 2 will be selected N / 6 times, etc.
We add them all: N / 6 * (1 + 2 + 3 + 4 + 5 + 6) = N / 6 * (21) = N * 21/6 = N * 3.5 ==> the average period for triggering N events was 3.5, which is more or less the maximum duration of the timer.
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