# Stats

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The time you wait in a bank until you are served by a teller is exponentially distributed with mean 10 minutes. Calculate the probability that you will have to wait between 5 and 15 minutes.

• Stats -

X is exponentially distributed, so the probability density function f(x) = m.exp(-m.x), where E(X) = 1/m. You're told that the expected value is 10 mins, so 1/m = 10, so m = 1/10 = 0.1. So what you need to calculate is the integral of the pdf between 5 mins and 15 mins, that is, the integral of 0.1 times exp(-x/10) between those limits. That is -exp(-1.5) + exp(-0.5), which is approximately 0.383. So the probability of your having to wait between 5 and 15 minutes is 0.383. (I think I've got that right, but do check my reasoning.)

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