Spread of the wait between trades (exponential)

Question

On a quiet name, the time between consecutive trades is exponential with rate \lambda = 0.5 per minute. What are the mean and variance of the waiting time between two trades?

Accepted Answer

For an exponential with rate \lambda (meaning \lambda events per unit time on average): - Mean wait = 1/\lambda. A rate of 0.5 trades per minute means one trade every 2 minutes on average. - Variance = 1/\lambda^2, so the standard deviation is also 1/\lambda. Here 1/\lambda = 1/0.5 = 2 minutes, so 	ext{mean} = 2 	ext{ min}, \qquad \operatorname{Var} = \frac{1}{0.5^2} = 4 	ext{ min}^2. A quirk of the exponential: the standard deviation (\sqrt{4} = 2) equals the mean. Waits are as variable as they are long, which is why exponential gaps feel bursty. Takeaway. An exponential with rate \lambda…

Spread of the wait between trades (exponential)

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