Average end point of 1-dimensional random walk?

Question

Is it possible to estimate the average end point of a 1-dimensional random walk of n steps where the probability of going "left" is p and going "right" is 1-p?

Thanks.

Answer 1

Let $X_t$ be the position at time $t$.

We can think of the random walk as being the sum of a set of random variables, $Z_1$, $Z_2$, $Z_3$, ..., each taking the value 1 with probability $p$ and the value -1 with probability $1-p$.

Hence:

$$X_t = \sum\limits_{i=1}^t Z_i$$

So by the linearity of expectation:

$$\mathbb{E}[X_t] = \sum\limits_{i=1}^t\mathbb{E}[Z_i] = \sum\limits_{i=1}^tp -(1-p) = (2p - 1)t$$