For the random walk with step sizes:
$S_i = \begin{cases}
&+1 &\text{probability} &p, \\
&-2 &\text{probability} &q=1-p
\end{cases}$
Let $T_n = \sum_{i=1}^mS_i$ be the displacement after a fixed, not random, number of steps $n$.
Find the probability distribution: $P(T_n=t)$
and the mean and variance of $T_n$ in terms of a general $n$ and $p$.
Best Answer
Note that $\frac{S_i+2}{3}$ is distributed as $B(1,p)$, giving that $\frac{T_n+2n}{3}$ is distributed as $B(n,p)$. Thus:
$$\mathbb{E}\left(\frac{T_n+2n}{3}\right)=np, \mathbb{V}\left(\frac{T_n+2n}{3}\right)=np(1-p)$$
from which you can derive:
$$\mathbb{E}(T_n)=n(3p-2), \mathbb{V}(T_n)=9np(1-p)$$
Furthermore, knowledge of the binomial distribution gives that:
$$\mathbb{P}\left(\frac{T_n+2n}{3}=k\right)=\binom{n}{k}p^k(1-p)^{n-k}$$
which can be manipulated to find the distribution of $T_n$ itself.