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Half-logistic distribution

From Wikipedia, the free encyclopedia

Half-logistic distribution
Probability density function
Probability density plots of half-logistic distribution
Cumulative distribution function
Cumulative distribution plots of half-logistic distribution
Support
PDF
CDF
Mean
Median
Mode 0
Variance

In probability theory and statistics, the half-logistic distribution is a continuous probability distribution—the distribution of the absolute value of a random variable following the logistic distribution. That is, for

where Y is a logistic random variable, X is a half-logistic random variable.

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  • ✪ Why the US' Land is Blocky
  • ✪ Lecture 23: Beta distribution | Statistics 110
  • ✪ Mean and Variance of Normal Distribution
  • ✪ Brazil's Geography Problem
  • ✪ An Introduction to the Poisson Distribution

Transcription

This video was made possible by Mozilla. Learn which products respect your privacy with their “Privacy Not Included" buyers guide at the link in the description. If you’ve ever flown over this US, it probably looked like this. If you’ve ever flown over Europe or pretty much anywhere else, it probably looked like this. Although that’s too much of a generalization, because if you fly over these eastern states, most of Texas, or parts of New Mexico it probably looked like this while if you flew over Louisiana, parts of Michigan, or parts of Wisconsin it probably looked like this. The fast answer to why this is is because all of these places had significant settlements before 1785 when Congress approved this—the Land Ordinance of 1785. A document people cared so little about that they didn’t even make a second version after their edits. In what was the British part of North America, land distribution was done with little organization so things look like this and this and this. They may try to disguise it with their recreational queuing but Britain loved chaos. Borders between properties were just arbitrarily set. When the US defeated Britain in the revolutionary war the two countries signed the treaty of Paris—wait, not that one, not that one, wow Paris needs to chill with these treaties—the 1783 treaty of Paris which officially ended the war between the two countries and set the boundaries between the British part of America and the United States. Through that, Britain gave away all this land to the US that it now had to divide up. And divide it did. Most of the US is divided into thousands upon thousands of townships. Townships are six miles by six miles and therefore 36 square miles. Each of these square miles is a section and the 36 sections are numbered 1 to 36. For the most part, the US sold these sections off to settlers with the exception of sections 8, 11, 16, 26, and 29. Section 16 was always reserved for education. In a lot of cases schools were physically put in section 16 of their township, so you’ll still find many schools today in section 16, but in many other cases the state just sold of the section 16’s and the proceeds were put towards public education. The other reserved sections—8, 11, 26, and 29—were held by the federal government for later sale with the expectation that the land prices would increase. Every state was designated a baseline and meridian from which the numbering of the townships would begin. For example, St Francis, Kansas is in S22 T3S R40W since it’s in section 22 3 townships south of the baseline and 40 townships west of the meridian. Most places have renamed the different townships, but some, such as lovely T9 R14 in Maine, have not. Since Maine still has townships with populations near zero they just haven’t bothered to rename many so you’ll still see blocks of land labeled on maps using this system. All across the US you can see the effect of this blocky surveying system. Towns in some places like Kansas and Nebraska tend to be on average six miles apart so that there’s one per township and almost every road tends to lie between two sections. In many cases, the standardization of land sizes is actually quite useful. You might have noticed all these circles within squares when looking down from airplanes. Those have nothing to do with surveying patterns at all, they have to do with sprinklers. Farmers use these sprinklers on wheels that pivot around a central point to water their plants so, the plants grow just within the circle and therefore make this pattern. Farmers usually split one section into four quarters and then use one of these central pivot sprinklers and therefore, because each quarter is a half mile wide pretty much every central pivot sprinkler is designed to be exactly one quarter mile long. But why do these places not look square like the American parts or disorganized like the British parts. Or more specifically, why do these ones look like this. Basically, because France. Compare this image of the Dordogne in France to the Mississippi in Louisiana—they’re nearly identical. Since France originally settled Louisiana, they decided how the land would be allocated and they chose this style. These long tracts, called arpents, were designed to give the maximum number of people possible access to the rivers so they could move their crops by boat. The idea was also that everyone would live on the river-side of their property so that, unlike with the US system, everyone could live relatively close to each other in order to socialize and eat cheese and stuff. Therefore, still today, the farmland in Louisiana looks like this. But what looks like this is Mozilla’s “Privacy Not Included” buyers guide. The holidays are coming and a good gift is one that won’t spy on people, but the good news is that Mozilla has created a completely free and super-simple way to figure out how much different products respect people’s privacy. For example, you can learn that the Google Home shares data with third parties for advertising while Edwin the Duck tracks users locations. Instead of just hoping that you’ll be fine when a big data breach happens or wishing companies would respect your privacy more, make a difference with your wallet and buy the products that give you security. So, once again, when you’re buying gifts this holiday season, don’t forget to take a look at Mozilla’s “Privacy Not Included” buyers guide with the link in the description.

Contents

Specification

Cumulative distribution function

The cumulative distribution function (cdf) of the half-logistic distribution is intimately related to the cdf of the logistic distribution. Formally, if F(k) is the cdf for the logistic distribution, then G(k) = 2F(k) − 1 is the cdf of a half-logistic distribution. Specifically,

Probability density function

Similarly, the probability density function (pdf) of the half-logistic distribution is g(k) = 2f(k) if f(k) is the pdf of the logistic distribution. Explicitly,

References

  • Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994). "23.11". Continuous univariate distributions. 2 (2nd ed.). New York: Wiley. p. 150.
  • George, Olusegun; Meenakshi Devidas (1992). "Some Related Distributions". In N. Balakrishnan (ed.). Handbook of the Logistic Distribution. New York: Marcel Dekker, Inc. pp. 232–234. ISBN 0-8247-8587-8.
  • Olapade, A.K. (2003), "On characterizations of the half-logistic distribution" (PDF), InterStat, 2003 (February): 2, ISSN 1941-689X
This page was last edited on 8 November 2019, at 18:29
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