Comparing Normal and Poisson Distributions
You are modeling event counts, such as number of clicks, and continuous measurements, such as response time. You need to choose an appropriate distribution and understand when one can approximate the other.
Explain the main differences between the Normal and Poisson distributions, when a Poisson can be approximated by a Normal distribution, and how to derive the mean and variance for each.
Constraints & Assumptions
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Discuss support, parameters, shape, assumptions, and common use cases.
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Make clear that Poisson is discrete and Normal is continuous.
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Include approximation conditions and continuity correction.
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Derive or justify the mean and variance rather than only stating them.
Clarifying Questions to Ask
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Are the observations counts, rates, proportions, or continuous measurements?
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Are events independent and occurring at a roughly constant rate?
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Is the count mean large enough for a Normal approximation?
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Is there overdispersion or underdispersion relative to the Poisson assumption?
Part 1 - Distribution Differences
Compare the Normal and Poisson distributions.
What This Part Should Cover
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Poisson support is nonnegative integers; Normal support is all real numbers.
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Poisson has one rate parameter lambda with mean and variance both lambda.
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Normal has mean mu and variance sigma squared as separate parameters.
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Poisson is often right-skewed for small lambda; Normal is symmetric and bell-shaped.
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Poisson models counts in fixed intervals; Normal models continuous measurements or sums of many small effects.
Part 2 - Normal Approximation to Poisson
State when a Poisson distribution can be approximated by a Normal distribution.
What This Part Should Cover
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Use the approximation Poisson(lambda) is approximately Normal(lambda, lambda) when lambda is large.
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Give a rule of thumb such as lambda at least 10 or 20 depending on required accuracy.
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Use continuity correction for discrete probability ranges.
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Explain that the approximation is poor for small lambda or tail probabilities near zero.
Part 3 - Mean and Variance
Derive or justify the mean and variance for each distribution.
What This Part Should Cover
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State E[X] = lambda and Var(X) = lambda for Poisson.
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State E[X] = mu and Var(X) = sigma squared for Normal.
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Derive Poisson moments from the probability mass function, probability-generating function, or moment-generating function.
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Derive Normal moments from its standardization or moment-generating function.
Follow-up Questions
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What distribution would you use if count data are overdispersed?
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Why can a Normal model be problematic for small count outcomes?
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How would you model click counts with different exposure times?