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? Any help will be appreciated.
That specific exponent ensures that the characteristic function of a distribution always exists as well as having some nice properties. It is of course possible to take another exponent such as the moment generating function $$ M_X(t) = E[e^{tX}] $$ Which is not guaranteed to exist. Take for example the Cauchy distribution.
The exponent also guarantees nice properties of the characteristic function (Properties on the Wikipedia page). One in particular, is that the characteristic function is bounded $$ |\varphi_X(t)| \leq 1 $$
The Fourier transform is the complex conjugate of the characteristic function.