Expected value of a delta function

Question

I have a certain question concerning the expected value of a density function, which is given by a delta function: If $$ f(x)= \frac{k}{m}\cdot \delta_{q}(x)$$ defines a density function, with $k,m$ being constants and $$\delta_q(x)\left\{ \begin{array}{ll} \infty & ,\, x=q \\ 0 & ,\,\text{else} \\ \end{array} \right. $$ being the delta function at $q$. Then the expected value is given by $$\frac{k}{m}\int_{-\infty}^{\infty}x\cdot \delta_q(x)\, \text{d}x \,.$$

Is there any clever way to calculate this expected value? I don't really know how to handle the delta function. Should one use integration by parts?

Thank you very much for any help!

Answer 1

By the definition of the $\delta$ function, for all continous functions $\phi$:

$$\int_{-\infty}^\infty \phi(x)\delta_q(x)dx = \phi(q)$$

Thus, the expected value of $f$ is $\frac{k}{m}q$.