Eli5: how did statisticians conceptualise new distributions ( normal , t etc) and also give a mathematical formula for the same?
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In most cases, they are defined to have specific properties. Gauss originally defined the normal distribution as the smooth, symmetric distribution that maximises the probability of obtaining a given set of independent measurements of the same quantity, while having a single peak at the mean of those measurements. From there, it’s not particularly complicated to derive the formula (the most interesting part is the “Gaussian integral trick”, a weird technique that is used to obtain the factor of sqrt(pi) in the formula).
Often, there are multiple different ways of defining a given distribution. For example, you can also define the normal distribution in terms of the central limit theorem, or in terms of a limiting case of the binomial theorem. These lead to exactly the same formula. In fact, de Moivre obtained the formula for the normal distribution based on the latter before Gauss came up with it, but he didn’t interpret it as a probability distribution.
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