You have 3 points of data with the values 1, 2 and 3. Which distribution is most likely?

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BlackTom AI · Accepted Answer

The question provides three data points: 1, 2, and 3, and asks which distribution is most likely.
First, observe that there are no answer choices to evaluate, so we cannot compare options directly. Instead, we reason from the data-generating process and common modeling assumptions.
If we assume the data come from a normal (Gaussian) distribution, a reasonable estimate for the mean mu is the sample mean. The mean of 1, 2, 3 is (1+2+3)/3 = 2, so mu ≈ 2.
For the standard deviation sigma, there are two common estimates:
- Maximum likelihood estimate (MLE) uses the population variance formula with n in the denominator. Here, the squared deviations from the mean are (1-2)^2 = 1, (2-2)^2 = 0, (3-2)^2 = 1, totaling 2. Dividing by n=3 gives variance 2/3, so sigma ≈ sqrt(2/3) ≈ 0.816.
- Unbiased (sample) estimate uses (n-1) in the denominator. That yields variance 2/(3-1) = 1, so sigma ≈ 1.
Thus, under a normal-model assumption, a plausible distribution would be N(mu=2, sigma≈0.82) or N(mu=2, sigma≈1), depending on whether one uses the MLE or unbiased estimator for sigma.
Now, consider the provided option in the prompt: N(mu=8, sigma=2). This specifies a mean of 8, which is vastly inconsistent with the observed data (the data are centered around 2). Such a distribution would assign most probability mass far from the observed values, making it an unlikely model for these three points.
Because there are no alternative options to compare, we cannot identify another distribution to contrast. However, any distribution with a mean near 2 (and a reasonable sigma in the vicinity of 1 or less) would be far more compatible with the data than mu=8. Conversely, if one were to assume a different family (e.g., Poisson, exponential), the data 1, 2, 3 could be plausible under a Poisson with lambda ≈ 2, but the prompt explicitly asks which distribution is most likely and often a normal model is assumed unless stated otherwise.
In summary, the data 1, 2, 3 imply a central tendency around 2, not 8, so N(mu=8, sigma=2) is not a plausible fit for these observations under common normal-model assumptions. The most reasonable normal-model descriptions would center mu at 2 with sigma in the range of about 0.8 to 1, depending on the estimator used.

ERMCPS5350_005_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT Class 9 Practice Question

You have 3 points of data with the values 1, 2 and 3. Which distribution is most likely?

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