Email Arrivals 

Assuming email arrivals follow a PoissonProcess with a rate of 25 per hour, find the probability that you get no mail for a 15-minute period.

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X emailArrivals = PoissonProcess[25];

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X Probability[x[0.25] == 0, x \[Distributed] emailArrivals]

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Find the expected number of emails over time.

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X Mean[emailArrivals[t]]

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Find a 95% confidence band for the expected number of emails.

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X Plot[{Mean[emailArrivals[t]], Quantile[emailArrivals[t], 0.025], Quantile[emailArrivals[t], 0.975]}, {t, 0, 8}, Filling -> {2 -> {3}}]

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Simulate the email arrival process for 10 different one-hour periods.

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X RandomFunction[emailArrivals, {0, 1}, 10]

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X RandomFunction[emailArrivals, {0, 1}, 10]; ListLinePlot[%, InterpolationOrder -> 0]

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