The definition of MGF of a random variable with PDF f(x) is

E[e^{sX}] = int_{-inf}^{inf} f(x) e^{sx} dx

The definition of Laplace Transform of a signal f(t) is

F(s) = int _{-inf}^{inf} f(t) e^{-st} dt

Hence MGF is 'flipped' Laplace transform

Now for we know that the MGF of sum independent RVs is the product of their MGFs. So if we take the inverse Laplace transform, the density of the sum is convolution of the individual densities.

Similarly, if we take derivative in frequency domain, that is same as multiplying in time domain: So M'_X(s) is the 'flipped Laplace transform' of x f(x) and its value at s=0 is the 'DC-gain' of the signal.

And so on... the properties are all immediate consequence of the definition of MGF and since the definition is essentially the same as that of a Laplace transform , there is an equivalent property in signals and systems as well.

The most honest source is Voja's My Deep Guide Channel on YouTube. Very detailed and informative reviews.

Comments URL: https://news.ycombinator.com/item?id=38797598

Points: 5

# Comments: 0

\begin{conccurrent}{Hacker News Thread} I'm writing in an HN thread. \end{concurrent}

is not so different from the markdown source. All the benefits you mention are simply from using semantic markup.