Does the Dirac delta distribution have a mean?

Question

Assuming that the Dirac delta distribution has a mean, it can be nothing else, but 0. But I'm not sure if it makes sense to say that.

Answer 1

Using the sifting property of $\delta(x)$, the mean of a 'random variable' with dirac delta distribution is $0$. In particular, $$\int_{-\infty}^{\infty} x \delta(x) dx = 0.$$ In general, the mean of a 'random variable' with shifted dirac delta PDF is $$\int_{-\infty}^{\infty} x \delta(x-a) dx = a.$$

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Note that when $f_{X}(x) = \delta(x-a)$ then $\Pr(X=a)=1$ and $\Pr(X\neq a)=0$.

Also, the sifting property of $\delta(x)$ is $\int_{-\infty}^{\infty} g(x)\delta(x-a) dx = g(a)$.