Given a function $f$ with a range between $[0,1]$, and a point $x$ that walks randomly on the number line, initially at the zero point of the number line. $x$ will move $1$ unit to the left or right, and the probability of moving right is $f(x)$. Where is the expected $x$ line after walking $t$ times?
Here's an example: $f(x)=\frac{1}{e^x+1}$. $x$ is always expected to be at zero on the number line.
If the solution to the problem does not exist or it is difficult to find a solution, then you can answer these special cases first: $$ f(x)=\frac{1}{2}(1+\sin x);\\ f(x)=\frac{1}{1+\sqrt{\pi}e^{\frac{1}{\pi(1-\ln (\frac{1}{x}+1))}}} $$.
I try to look for patterns in the special cases (the cases I give). By setting the $t=10$, my calculation is that the expectation of the $f(x)=\frac{1}{2}(1+\sin x)$ in my given case is close to $0$, and the expectation of the $f(x)=\frac{1}{1+\sqrt{\pi}e^{\frac{1}{\pi(1-\ln (\frac{1}{x}+1))}}}$ is close to $-3.67$.Procedural simulations seem to indicate that the distance of the random walk is proportional to time, but I don't know why.