and convincing ideas: its very success tempted people to apply it in realms for which it was ill adapted, sciences far removed from its native astronomy: chemistry, biology, the ever hoped-for social sciences. Here, phenomena did not necessarily fall neatly in a normal distribution around a âtrueâ value. New patterns, new distributions appeared that required a new calculus.
One early example of the new curves made necessary by wider observations was named for Laplaceâs student, Poisson, a former law clerk who hoped to apply probability to evidence and testimony. Poisson identified a class of events that, like crime, could happen quite easily at any time, but in fact happen rarely. Almost anyone in the world, for instance, could call you on the telephone right nowâbut itâs highly unlikely that a ring actually coincided with the moment your eye passed that dash. Quite a lot of human affairs turn out to be like this: the chance of being hit by a car in Rome on any one day is very small, although passing a lifetime there makes the likelihood of at least a bump quite high. You might have a real interest in knowing the relative likelihoods of being run into once, twice, or more times.
Poissonâs distribution is most like real life in not supposing that we know the actual probability of an event in advance. We already âknowâ the probabilities attached to each side of a die; we âknowâ the laws that should govern our observation of a planetâs path. We can therefore multiply that known probability by the number of trials to give us our probability distribution. But in real life, we may not have that information; all we have is the product of the multiplication: the number of things that actually happened.
Poissonâs curve, more a steeple than a bell, plots this product of probability times number of trials for things that happen rarely but have many opportunities to happen. The classical case of a Poisson distribution, studied and presented by the Russian-Polish statistician Ladislaus Bortkiewicz, is the number of cavalry troopers kicked to death by horses in 14 corps of the German army between 1875 and 1894.
Here are the raw figures:
The total of trials (20 years x 14 corps) is 280; the total of deaths is 196; the figure for deaths per trial, therefore, is 0.7.
The Poisson formula for this would be
where m is the number of deaths per year whose probability we want to gauge. If m = 1, the probability is 0.3476. If this is applied to 280 experiments, the probable number of times one death would occur in any corps during one year is 97.3 (in fact, it was 91). The theoretical distribution is remarkably close to the actual one (which is probably why this is cited as the classical case):
If it happens you are not a cavalryman, what use could you make of this? Perhaps the best characterization of Poissonâs distribution is that, whereas the normal distribution covers anticipated events, Poissonâs covers events that are feared or hoped for (or both, as in the case of telephone calls). Supermarkets use it to predict the likelihood of running out of a given item on a given day; power companies the likelihood of a surge in demand. It also governed the chance that any one part of south London would be hit by a V2 rocket in 1944.
If you live in a large city, you might consider Poissonâs distribution as governing your hope of meeting the love of your life. This suggests some interesting conclusions. Woody Allen pointed out that being bisexual doubles oneâs chance of a date on Saturday night; but sadly Poissonâs curve shows very little change in response to even a doubling of innate probability, since that is still very small compared with the vast number of trials. Your chance of fulfillment remains dispiritingly low. Encouragingly, however, the greatest proportion of probability remains packed in the middle of the curve, implying that your best chance comes
One early example of the new curves made necessary by wider observations was named for Laplaceâs student, Poisson, a former law clerk who hoped to apply probability to evidence and testimony. Poisson identified a class of events that, like crime, could happen quite easily at any time, but in fact happen rarely. Almost anyone in the world, for instance, could call you on the telephone right nowâbut itâs highly unlikely that a ring actually coincided with the moment your eye passed that dash. Quite a lot of human affairs turn out to be like this: the chance of being hit by a car in Rome on any one day is very small, although passing a lifetime there makes the likelihood of at least a bump quite high. You might have a real interest in knowing the relative likelihoods of being run into once, twice, or more times.
Poissonâs distribution is most like real life in not supposing that we know the actual probability of an event in advance. We already âknowâ the probabilities attached to each side of a die; we âknowâ the laws that should govern our observation of a planetâs path. We can therefore multiply that known probability by the number of trials to give us our probability distribution. But in real life, we may not have that information; all we have is the product of the multiplication: the number of things that actually happened.
Poissonâs curve, more a steeple than a bell, plots this product of probability times number of trials for things that happen rarely but have many opportunities to happen. The classical case of a Poisson distribution, studied and presented by the Russian-Polish statistician Ladislaus Bortkiewicz, is the number of cavalry troopers kicked to death by horses in 14 corps of the German army between 1875 and 1894.
Here are the raw figures:
The total of trials (20 years x 14 corps) is 280; the total of deaths is 196; the figure for deaths per trial, therefore, is 0.7.
The Poisson formula for this would be
where m is the number of deaths per year whose probability we want to gauge. If m = 1, the probability is 0.3476. If this is applied to 280 experiments, the probable number of times one death would occur in any corps during one year is 97.3 (in fact, it was 91). The theoretical distribution is remarkably close to the actual one (which is probably why this is cited as the classical case):
If it happens you are not a cavalryman, what use could you make of this? Perhaps the best characterization of Poissonâs distribution is that, whereas the normal distribution covers anticipated events, Poissonâs covers events that are feared or hoped for (or both, as in the case of telephone calls). Supermarkets use it to predict the likelihood of running out of a given item on a given day; power companies the likelihood of a surge in demand. It also governed the chance that any one part of south London would be hit by a V2 rocket in 1944.
If you live in a large city, you might consider Poissonâs distribution as governing your hope of meeting the love of your life. This suggests some interesting conclusions. Woody Allen pointed out that being bisexual doubles oneâs chance of a date on Saturday night; but sadly Poissonâs curve shows very little change in response to even a doubling of innate probability, since that is still very small compared with the vast number of trials. Your chance of fulfillment remains dispiritingly low. Encouragingly, however, the greatest proportion of probability remains packed in the middle of the curve, implying that your best chance comes
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