In statistics, Moment Generating Function (MGF) is used to decompose the probability distribution to its moments. The resultant MGF when expanded into a series (a discrete approach) or split by various order derivatives (a continuous approach) results into the various moments. Thanks to this usage, when I was introduced to Laplace transforms in a separate mathematics course, I understood the transform not just as a mathematical tool but also its purpose in decomposing the moments.
FT was a step further away, it was a cryptic puzzle to start. By the name I understood that it decomposes something, but failed to see what. The lack of internet resources made it difficult. I rummaged the library on FT but failed to come up with anything meaningful as they all dealt with solving functions rather than explaining why it was used in the first place. It was a passing remark in a textbook that told me that it decomposes signals. The decomposition into sine and cosine waves became immediately apparent.
I still wonder why no textbook ever explain what a mathematical tool is, but just start presenting functions. There are few good teachers who help you understand the fundamentals but none seem to ever write a textbook.
I might have gotten through FT, but stochastic processes continue to haunt me, especially when they're so erroneously put to use. Like most in the field, I can solve stochastic differential equations, find resulting distributions using analytical or numerical approaches. I also have the ability to tell when the results are meaningless (a huge bonus in my field). But yet I'm not endowed with ability to build the fundamental processes (that result in meaningless outcomes :P). Wonder where I'll ever learn to build the fundamental processes in their true form. Is there a teacher somewhere?
