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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, ...


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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 ...


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