An SRE team is analyzing the daily count of service outages for a cloud platform. Over the last 365 days the observed frequencies are: 0 outages on 310 days, 1 outage on 45 days, and 2 outages on 10 days (no day had more than two outages). The sample mean is 0.18 outages per day and the sample variance is 0.20. To develop a generative model for the number of outages per day, which distribution and supporting rationale provides the most statistically appropriate starting point?
Poisson distribution - the near-equality of the sample mean and variance supports a Poisson rate parameter λ ≈ 0.18 for rare, independent daily outages.
Binomial distribution - because the count of outages can be viewed as successes in 365 daily trials with variance np(1 − p).
Student's t-distribution - its heavier tails better model the occasional two-outage days in a small sample.
Power law distribution - heavy-tailed behavior explains low-probability, high-impact outage counts.
The Poisson distribution is designed for modeling the number of independent events that occur in a fixed interval when those events are rare and occur at a constant average rate. A defining property of the Poisson distribution is that its mean and variance are both equal to the rate parameter λ. Because the observed data are non-negative integer counts, the mean (0.18) is very close to the variance (0.20), and outages are presumed independent from day to day, the Poisson distribution is the most appropriate first model.
The binomial distribution is inappropriate here because it requires a fixed number of identical trials (n) each day; in this context n would need to be the unknown number of "possible outage opportunities" within a day, and its variance is np(1 − p), which is strictly less than the mean np. Since the sample variance is slightly greater than the sample mean, the binomial is a poorer fit than the Poisson.
The Student's t-distribution is a continuous distribution used for inference on sample means when population variance is unknown; it cannot generate non-negative integer counts.
A power-law distribution is heavy-tailed and continuous or defined on positive integers with probabilities that decay polynomially; it is suited to modeling extreme events across many orders of magnitude, not tightly bounded low counts such as 0, 1, 2 outages. Therefore, only the Poisson model aligns with both the empirical moment relationship and the data-generation mechanism.
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Why is the binomial distribution not suitable for modeling daily outages?
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What differentiates the Poisson distribution from heavy-tailed distributions like the power law?