A student who is attending probability 101, learned about normal distribution and generating functions recently.

We are given a "generating function" as follows: $$G(t)=<e^{itx}>=\int_{-\infty}^{+\infty}e^{itx}p(x)dx$$

And we are asked to explain the following:

  1. why is there an exponent

  2. why do we use the imaginary unit

As I am really new to probability, few stuff that I knew is the function is a characteristic function, and it somehow related to Fourier transform (which I haven't learned yet). Could someone show me what to search for, or try to explain the above questions?


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