• Many books and classes in probability mention some mysterious entities called “Characteristic functions”, with little or no motivation. For students like myself, concepts without proper motivation and excitement do not register in the brain, that’s why I almost completely forgot all I’ve learned about characteristic functions from my first classes/books. The truth is that characteristic functions turn out to be incredibly useful in probability and play an important reason for the following four reasons. 1) Characteristic function is the other side…

• Assume we have a zero-mean Gaussian RV $\mathbf x$ with variance $\sigma^2$, and we want to compute its moments, i.e., $E\{\mathbf x^n\}$. The moments for odd $n$ are zero because the density function of our RV $f(x)$ is an even function. But how to compute the moments. i.e., $$E\{\mathbf x^n\} =\int\limits_{-\infty}^{\infty} x^n f(x) dx =\frac{1}{\sigma \sqrt{2\pi}}\int\limits_{-\infty}^{\infty} x^n e^{-x^2/2\sigma^2} dx$$ for even $n$? Of course this is not a very difficult integral (can be computed through integration by parts etc.), but…