Can someone help me with this question? I have searched everywhere but cannot find how to answer this.
Consider a random walk in discrete time, $t=0,1,2, \ldots,$ and on a set of discrete sites $i=0,1,2, \ldots$ The walker starts from $i=0$. In each time step, the walker hops to the next site $(i \rightarrow i+1)$ with probability $q ;$ with probability $1-q$ the walker stays at its current location. The walker cannot move backward. State the probability that the walker is at site $i$ at time $t .$ What is the expected position $\langle i\rangle$ at time $t ?$ (A detailed calculation is not required.)
Think of it like flipping a (biased) coin $t$ times. Suppose you are interested in $t=2$. If you flip two coins, then you could either get HH, HT, TH, TT. Heads heads corresponds to going forward by $2$, HT and TH by $1$, etc. The probabilities of each are $P(i =2) = q^2$, $P(i = 1) = 2 q(1-q)$ (two ways to get 1 head), and $P(i = 0) = (1-q)^2$. Then extend this argument to more flips - you may recall the binomial distribution.