For example, a fair coin would have a sample space = {H, T}. A particular probability function could take on different probability values (in the interval $[0,1]$) relative to time, such function could be $\mathbb{P}(H) =|\sin{(t)}|$ with $t \in \mathbf{R}^+$, with of course $\mathbb{P}(T) = 1-|\sin{(t)}|$. I chose $|\sin{(t)}|$ so as for the function to lie between $0$ and $1$.
Supposing one picks, in intervals of time $n$ where $n= t_{1}-t_{0}$ between H or T with a probability equal to $\mathbb{P}(H) =|\sin{t}|$ and were to move up the $y$ axis by $1$ when H occurred, and $-1$ when T occurred (in intervals of n) what would be the function that links the continously changing probability function $\mathbb{P}(H) =|\sin{(t)}|$ with the interval n function, defined only at points n, so as to create a discrete function that maps the path of the coin tosses.
In other words, how does one create a discrete function where the probability of an outcome continuously changes with respect to time?
I apologize for the terribly explained question, i have tried my best, i will gladly answer any,very comprehensible, confusion.
Define:
$X_n = 1,$ with probability $p_n = |sin(n)|$
$X_n = -1,$ with probability $1-p_n = 1 - |sin(n)|$
where $n = 1, 2, 3, ...$
Are you looking for something like this?