Lesson plan
Split normal distribution
In probability theory and statistics, the split normal distribution also known as the two-piece normal distribution results from joining at the mode the corresponding halves of two normal distributions with the same mode but different variances. It is claimed by Johnson et al. that this distribution was introduced by Gibbons and Mylroie and by John. But these are two of several independent rediscoveries of the Zweiseitige Gauss'sche Gesetz introduced in the posthumously published Kollektivmasslehre (1897) of Gustav Theodor Fechner (1801-1887), see Wallis (2014). Another rediscovery has appeared more recently in a finance journal. {{Probability distribution | name = Split-normal | type = density | notation = \mathcal{SN}(\mu,\,\sigma_1,\sigma_2) | parameters = \mu \in \Re — mode (location, real) \sigma_1 > 0 — left-hand-side standard deviation (scale, real) \sigma_2 > 0 — right-hand-side standard deviation (scale, real) | support = x \in \Re | | pdf = A \exp \left(- \frac {(x-\mu)^2}{2 \sigma_1^2}\right) \quad \text{if } x A \exp \left(- \frac {(x-\mu)^2}{2 \sigma_2^2}\right) \quad \text{otherwise,} \text{where} \quad A= \sqrt{2/\pi} (\sigma_1+\sigma_2)^{-1} | mode = \mu | mean = \mu+\sqrt{2 / \pi}(\sigma_2-\sigma_1) | variance = (1-2 / \pi)(\sigma_2-\sigma_1)^2 + \sigma_1 \sigma_2 | skewness = \gamma_3 = \sqrt{\frac{2}{\pi}}(\sigma_2-\sigma_1)\left[\left(\frac{4}{\pi}-1\right)(…
Article summary
In probability theory and statistics, the split normal distribution also known as the two-piece normal distribution results from joining at the mode the corresponding halves of two normal distributions with the same mode but different variances. It is claimed by Johnson et al. that this distribution was introduced by Gibbons and Mylroie and by John. But these are two of several independent rediscoveries of the Zweiseitige Gauss'sche Gesetz introduced in the posthumously published Kollektivmasslehre (1897) of Gustav Theodor Fechner (1801-1887), see Wallis (2014). Another rediscovery has appeared more recently in a finance journal. {{Probability distribution | name = Split-normal | type = density | notation = \mathcal{SN}(\mu,\,\sigma_1,\sigma_2) | parameters = \mu \in \Re — mode (location, real) \sigma_1 > 0 — left-hand-side standard deviation (scale, real) \sigma_2 > 0 — right-hand-side standard deviation (scale, real) | support = x \in \Re | | pdf = A \exp \left(- \frac {(x-\mu)^2}{2 \sigma_1^2}\right) \quad \text{if } x A \exp \left(- \frac {(x-\mu)^2}{2 \sigma_2^2}\right) \quad \text{otherwise,} \text{where} \quad A= \sqrt{2/\pi} (\sigma_1+\sigma_2)^{-1} | mode = \mu | mean = \mu+\sqrt{2 / \pi}(\sigma_2-\sigma_1) | variance = (1-2 / \pi)(\sigma_2-\sigma_1)^2 + \sigma_1 \sigma_2 | skewness = \gamma_3 = \sqrt{\frac{2}{\pi}}(\sigma_2-\sigma_1)\left[\left(\frac{4}{\pi}-1\right)(…
Key claims
- publication date: 1897 — source: Wikipedia · confidence 0.95
Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- The record states that the publication date is "1897". What evidence would count against this claim?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
- Wikipedia https://en.wikipedia.org/w/index.php?oldid=1374403628
Key claims and bibliography are generated from the same tracked claims and sources as the article page.