Yes, they are all related to the Normal (Gaussian) distribution, which is the distribution of a continuous random variable.
The first and second are effectively the same; it is the definition of the Normal distribution function where sigma is the standard deviation, m or u is the mean and x is the dependent variable. For whatever reason, there is a dx on the end of the first one, which presumably relates to somebody trying to integrate it.
The third eq is the probability density of the Normal distribution variable Z, where Z = (x - u) / sigma which is a handy transform.
The fourth eq is what you get if you put a2 = 2 [sigma]2, and presumably integrate. Don't know what that N is there for.
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