When did the concept of probability density explicitly appear in mathematics?

Question

I was reading the wikipedia article about the normal distribution and it's attribution by some historians of science to De Moivre(althought he lacked the concept of the probability density function) and I was wondering as to when was the probability density function defined.

Answer 1

The answer depends somewhat on what "explicitly" means. The appearance of continuous distributions, and their densities, is generally attributed to Simpson. Namely, his 1757 response to Bayes's critique of his 1755 letter on de Moivre's theory of errors. As Stigler writes in The History of Statistics:

"Simpson's 1757 republication of the letter incorporated one other change - three and a half pages of additional material that is frequently cited now as the first publication of a continuous error distribution... Simpson (1757, p. 71) proposed to consider the situation ''when the error admits of any value whatever, whole or broken, within the proposed limits, or when the result of each observation is supposed to be accurately known." His analysis, in the spirit of the time, consisted of viewing the interval of possible errors (AB in Figure 2.3) as being divided into an indefinitely large number of indefinitely small subintervals, and effectively passing to the limit in his earlier expression for the discrete case...

In modern terminology, Simpson's idea is that we may consider the triangle ABD as a probability density along the axis AB... the curve AFEFB represented the probability density for the mean error. He integrated the resulting density to find that the probability that the mean of t errors did not exceed $(1 - y/t)\times$CA in absolute value".

This said, Simpson is certainly very far from abstract notions of probability distribution and density. Bayes's 1764 essay is noted as a step forward, "remarkable for its careful treatment of probability densities, treating all probabilities as areas and not relying (in the later manner of both French and English mathematicians) on the interpretation of densities as attaching infinitesimal probability to points" (as Maxwell did as late as 1860). But he still deals with a specific density function. Finally, Laplace wrote a number of memoirs on the "error curve" in 1770s, where the said curve (density) gets denoted generally $\phi(x)$. He was aiming at the normal distribution density, but did not succeed at identifying it at the time, and had to treat his $\phi(x)$ more abstractly:

"All early workers, from Simpson on, accepted it as given that the curve should be symmetrical and that the chance of an error should decrease toward zero as the magnitude of the error increased. Laplace himself had repeated these conditions in his 1774 memoir, as I have noted. The problem was that there were too many possibilities, and the choice of just one was critical to the mean to be obtained. As Laplace wrote, "But of an infinite number of possible functions, which choice is to be preferred?"

Earlier writers had confronted the choice of an error curve by making an arbitrary selection. Simpson, for example, had based his choice of the uniform and triangular distributions upon mathematical expediency. For his limited aim (showing that an arithmetic mean was better than a single observation) this arbitrariness was not a major drawback - his audience only needed to accept the distribution as qualitatively correct in order to believe his qualitative conclusion - that averaging increased accuracy. Laplace, however, was attempting an exact computation, and he needed both a convincing case for any choice of $\phi(x)$ he might make and a mathematical analysis equal to the task of calculating the best mean for that curve."

As for the terminology, it is quite recent. According to Earliest Known Uses of Some of the Words of Probability & Statistics Markoff came up with "Wahrscheinlichkeitsdichte" (probability density) only in 1912, and the "distribution function", of which it it the derivative, only appears in von Mises's Grundlagen der Wahrscheinlichkeitsrechnung (1919).

Answer 2

Hans Fischer, in his A History of the Central Limit Theorem writes:

The history of the CLT as a universal law begins with Laplace; all relevant studies in the 18th century, starting with [de Moivre 1733], essentially contained only approximations of the binominal distribution and their scope of application remained narrow. Laplace’s finding of 1810, according to which the additive coaction of a large number of independent random variables generally leads to probabilities that can, at least approximately, be calculated according to the normal distribution, substantially expanded the numerical possibilities of probability theory, especially in the discussion of mass phenomena

and

In one of his first published papers [in 1776], Laplace had already set out to determine the probability that the sum of the angles of inclination of comet orbits (or the arithmetic mean of these angles respectively) is within given limits. He assumed that all angles, which had to be measured against the ecliptic, were distributed randomly according to a uniform distribution between 0 and 90 degrees (and also tacitly presupposed that all angles were stochastically independent). Laplace succeeded in calculating these probabilities for an arbitrary number of comets via induction (with a minor mistake which was subsequently corrected in his paper of 1781) . In this [latter] paper, Laplace even introduced a general — however very intricate — method, based on convolutions of density functions, in order to exactly determine the probability that a sum of independent random variables (“quantités variables,” as Laplace put it) was within given limits

Personally, I feel that the notion of a probability density would have had earlier provenance; since, after the integral calculus had been invented, it wouldn't taken much to realise that the generalisation from finite probability to a continuous probability would require integration over a density (and the term density suggests that the notion migrated from physical theory).

It's probably worth adding that Kolmolgorov introduced the notion of a probability density in measure theoretic terms (the now standard notion) in his book, The Foundations of the Theory of Probability, first published in 1933.