In certain situations, it may be possible to use the __________ distribution to approximate the probability associated with a discrete distribution, like the binomial or Poisson distribution.

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BlackTom AI · Accepted Answer

The question asks about a distribution that can be used to approximate the probability of a discrete distribution, such as the binomial or Poisson, in certain situations.

Because the answer options are not provided in the prompt, I will explain the key idea and why the Normal distribution is typically chosen for approximation in these contexts.

In many practical problems, when the sample size is large or the underlying mean is large, the sum or average of many independent random contributions tends to follow a bell-shaped, continuous distribution rather than a discrete one. This is a consequence of the central limit theorem, which states that the sum (or average) of a large number of independent, identically distributed random variables with finite variance tends toward a Normal distribution, regardless of the original distribution.

For the binomial distribution, which counts the number of successes in n independent trials with probability p of success, the Normal distribution can be used as an approximation when both np and n(1-p) are sufficiently large (commonly np ≥ 5 and n(1-p) ≥ 5). In practice, a continuity correction is often applied to improve the approximation for discrete data.

Similarly, for the Poisson distribution with a large mean λ, the Normal distribution with mean and variance λ can serve as an approximation, especially when λ is not too small, again often with a continuity correction for better accuracy.

If the question had included alternative options (for example, using a Poisson or a skewed distribution as an approximation in different regimes), we would evaluate them against the conditions under which the Normal approximation is appropriate. The Normal choice is specifically tied to the central limit phenomenon and the requirement that the aggregate of many independent contributions behaves like a symmetric, continuous bell curve rather than a discrete count.

In summary, the Normal distribution is used to approximate discrete distributions like binomial or Poisson in scenarios with large n (and appropriate np and n(1-p)) or large λ, leveraging the central limit theorem and often applying a continuity correction to improve accuracy.

ECON 3640-090 Spring 2025 Lecture #16 Warm-Up

In certain situations, it may be possible to use the __________ distribution to approximate the probability associated with a discrete distribution, like the binomial or Poisson distribution.

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