2
$\begingroup$

As we know, the concept of a probability density function lets us identify a discrete probability (e.g. binomial) value with a corresponding area under a continuous curve (like the normal curve). De Moivre, however, did not have that concept (i.e. of a p.d.f.); nevertheless, after approximating the binomial probability (mass) function with the (normal) exponential function, he approximated the finite summation representing, say P(a x b), by the corresponding integral of the approximating exponential function, from x = a to x = b. In other words, he identified area with probability over a range of values of the binomial variable. We find this easy to swallow, today, because we can identify the ordinate value P(x = r) with a "skinny" rectangle having unit base. Thus, for us, probability is an area. But De Moivre never makes this association (that I can find). In fact, in 1730 he explicitly referred to a probability value as an ordinate (not an area) when he wrote (in Latin) "if the terms of the binomial are thought of as set upright, equally spaced at right angles to and above a straight line, the extremities of the terms follow a curve. The curve so described has two inflection points, one on each side of the maximal term." (see Stephen Sigler p. 76). I see no hint that De Moivre sees a histogram (vs. a series of perpendicular straight line segments drawn from the horizontal axis up to a point of his curve), so how did he get the idea of identifying area with probability?

  • 0
    good question! I know nothing of De Moivre, so I can't answer. I would speculate that mathematicians going that far back had the general idea that an integral is a continuum limit of a sum in their toolkit, although the formal idea of the Riemann/Darboux integral obviously came over 100 years later. Note DeMoivre didn't need to conceive of probability as area to do what you said... in his mind he could have merely been approximating a sum. – 2017-01-08
  • 0
    But "the terms of the binomial are thought of as set upright, equally spaced at right angles to and above a straight line" exactly describes a histogram, and "the extremities of the terms follow a curve" exactly alludes to the graph of a continuous function approximating the histogram, no? – 2017-01-08
  • 0
    This is what is still bothering me: – 2017-01-08
  • 0
    Also, I don't agree that De Moivre's wording, about the extremities of the terms following a curve, describe a histogram; the extremities of histogram components end in short horizontal line segments, not points. Line segments end in points, which then determine a curve. – 2017-01-08
  • 0
    (1) De Moivre's integration (in Corollary 2.) doesn't suggest a limit: he writes the expon'l f'n as an infinite series, then equates a certain probability to a new series, the term-by-term integral of the first series. He evaluates 7 terms of this latter series to estimate the probability in question. (2) In Corollary 6, the integral series doesn't converge fast enough "for which reason I make use in this case of the Artifice of Mechanic Quadratures,..; it consists in determining the Area of a Curve nearly, from.. a certain number of its Ordinates.." So his probability is area; why? – 2017-01-08
  • 0
    A binomial histogram suggests interpreting probability as area (of the histogram bars). DeMoivre did not have the histogram idea: he erected line segments which don't have area. But then, when he approximated the binomial function with the exponential curve, he approximated probability, say between 2 values of the binomial variable, by the area under the normal curve between those same 2 values. How did he get this idea? Was it already "out there" via someone else's work? This was in 1733, before the concept of a p.d.f. Thanks for your help. – 2017-01-09

0 Answers 0