Total logged errors over a day

Question

A trading system logs errors as a Poisson stream averaging 0.5 per hour. Errors in different hours are independent. Let $E$ be the total number of errors logged over a 24-hour day.

What distribution does $E$ follow, and what are its mean and variance?

Accepted Answer

Poisson counts over disjoint, independent time windows add, and the rate adds too. Over 24 hours at 0.5 per hour, the day-long rate is 24 	imes 0.5 = 12: E \sim 	ext{Poisson}(12), \qquad \mathbb{E}[E] = 12, \qquad \operatorname{Var}(E) = 12. Standard deviation \sqrt{12} \approx 3.46. Stretching the observation window from one hour to a day just scales the Poisson rate by the number of hours. Takeaway. Over 24 independent hours at rate 0.5, the total is Poisson(12), with mean and variance both 12.

Total logged errors over a day

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