Start by clearly defining both distributions, emphasizing their support and parameters, then derive the mean and variance for each using first principles. Finally, explain the conditions under which a Poisson can be approximated by a Normal, including the role of λ and continuity correction.
Pro tip: Mention that the Poisson is a discrete distribution often used for count data, while the Normal is continuous and arises from the Central Limit Theorem; this shows you understand when each is appropriate in practice.
State that the Normal is a continuous distribution defined on (-∞, ∞) with parameters μ and σ², while the Poisson is a discrete distribution on non-negative integers with parameter λ.
For Normal, the mean is μ and variance is σ², which can be derived from the probability density function or by using the moment-generating function.
For Poisson, both mean and variance equal λ. Derive using the probability mass function and the definition of expectation, or via the moment-generating function.
The Poisson can be approximated by a Normal when λ is large (typically λ > 20 or 30). Use continuity correction for better accuracy, and note that the Normal approximation improves as λ increases due to the Central Limit Theorem.
Highlight that Normal is continuous and symmetric, while Poisson is discrete and skewed for small λ. Also note that Poisson models counts per unit time/space, while Normal models continuous measurements.
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