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Dirac delta

The Dirac delta is a function (of sorts) that helps us—amond other things—to define a constant probability distribution in the language of probability densities. The simplest way to think about it is as a limit: take a Gaussian distribution in one dimension and shrink the variance to 0.

As the variance gets closer and closer to zero, you get a taller and taller peak, centering all the probabilitly mass closer and closer to a single point. The “distribution” that this process converges to is the Dirac delta. A density function that is zero everywhere and $\infty$ at a single point.

The Dirac delta can be hard to work with. If you need it, an alternative is to use a Gaussian with some small variance $\bc{\sigma}$, and then to take the limit as $\bc{\sigma} \to 0$.