I'm working on an assignment and I am having difficulty with the following question:
A particle is moving in one direction, covering each second 1 meter of distance starting at the the time $t = 0$ at the origin $X_0 = 0$ and moving in positive direction. It is known that at each second $t = 1, 2, 3 . . .$ the particle can change direction will travel one meter per second in the opposite direction. Denote the particle position at time $t = 0, 1, 2, . . .$ by $X_t$ and assume that the probability of the random hit is $p = 0.3$ and that the hits occur independently of each other. Calculate the characteristic function $X_t(x)$ of the random variable $X_t$ for $t = 4$ and $x = 2$.
So far I've mapped out the probabilities of being at each point for $t=4$ and probability the velocity being either $1m/s$ or $-1m/s$ till $t=4$ but am unsure what I should do next. Any input would be appreciated.