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Joseph Louis Lagrange

b. Turin 25 January 1736
d. Paris 10 April 1813

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Distribution of the Sample Mean

The distribution of the mean of n observations subject to certain assumptions of the distribution of errors is taken up in "Memoire sur l'utilité de la méthode de prendre le milieu entre les résultats de plusiers observations; dans lequel on examine les avantages de cette méthode par le calcul des probabilités; et où l'on résoud différens problèmes relatif à cette matière." This was published in Volume V of Miscellanea Taurinensia for the years 1770-1773 on pages 167-232. The paper discusses ten problems for which we follow the outline of Todhunter.

Problem Description Related Work
I To find the probability that the mean will be exact when errors have a symmetric distribution on the values -1, 0, 1.
II To find the probability that the mean will fall within a given radius about its exact value.
III To find the probability that the mean will fall with prescribed limits.
IV To find the most probable error in the mean.
V To find the most probable erro in the mean when the errors have an arbitrary discrete distribution.
VI The distribution of errors is not known but estimated from observation. To estimate the probability that the relative frequencies do not differ from the true value by more than an assigned quantity.
VII To find the probability that the mean has a prescribed value and lies within prescribed limits when the distribution of errors is uniform on some interval about 0. De Moivre,
Simpson, Advantage of the Mean
VIII To find the probability that the mean lies within prescribed limits when the errors have a discrete triangular distribution on a symmetric interval about 0.
X To find the probability that the mean lies within prescribed limits when the errors have a continuous distribution on an interval about 0.
XI To find the probability that the mean lies within prescribed limits when the errors are distributed proportional to one arc of the cosine centered at 0.


Difference Equations

Lagrange produced three papers on the integration of finite difference equations. The first is "Sur l'integration d'une équation différentielle à différences finies qui contient la théorie des suites récurrentes." This was  published in Miscellanea Taurinensia, T. 1, 1759.

It was many years later that he returned to the subject in "Recherches sur les suites recurrentes dont les termes varient de plusieurs manieres différentes, ou sur l'integration des équations linéaires aux différences finies et partielles; et sur l'usage de ces équations dans la théorie des hazards." This appeared in Nouveaux Mémoires de l'Académie ... Berlin pp. 183-272 for the year 1775 and published in 1777. The portion of the memoir concerning the theory of chances is contained in the pages 240-272.  This latter section solves the problem of points for several players, the duration of games and also an urn problem as outlined below. The paper of  Jean Trembley, "Disquisitio Elementaris circa Calculum Probabilium" purports to offer elementary solutions of these problems.

Problem Description Related Work
I To find the probability that an event is brought forth at least b times among a trials. Moivre, p. 15 and p. 27
II To find the probability that one event is brought forth at least b times, another at least c among a trials when a third event is possible as well. Trembley, Disquisitio Elementaris circa Calculum Probabilium, Problem 8. 
III To find the probability to bring forth, in an undetermined number of trials, the second of two events b times before the first has arrived a times. 
IV As the third problem excepting that one event must occur c times before the other two occur a and b times respectively.. Montmort, Propositions XL and XLI.
Moivre, 2nd ed., Problem VI.
Laplace, (1773) Problems XIV and XV.
V To find the probability that an event will occur at least b times more than it does not occur. Moivre, Problem LXV.
VI To find the probability that an event will occur at least b times more or less by c than it does not occur. Montmort, p. 268
Moivre, Problems LXIII & LXVIII, p. 191
VII To find the distribution of black and white balls in a sequence of urns after transfers have been made from one urn to another of randomly drawn balls a given number of times. Daniel Bernoulli, Disquisitiones Analyticae de novo problemate conjecturali.

Finally, we have "Recherches sur plusieurs points d'analyse relatifs à différens endroits des Mémoires précédens."  This memoir appeared in the Mémoires de l'Académie Royale des Sciences et Belles Lettres of Berlin for the years 1792-3, published in 1798, pp. 247-57. It does not directly concern the theory of probability but rather extends the method of solution of difference equations.

Annuities

"Mémoire sur une question concernant les annuités," published in 1798 in the Mémoires de l'Académie Royale des Sciences et Belles Lettres of Berlin for the years 1792-3, pp. 235-246. This paper was read to the Academy ten years prior. This paper adds nothing new as Moivre had already solved its problem in his Treatise of Annuities on Lives.