> [!definition] > > A random walk $S(n)$ is a [[Stochastic Process|stochastic process]] based on a [[Random Sample|random sample]] of size $n$, where the [[Probability Distribution|probability distribution]] of the [[Random Variable|random variable]] $X$ is: > $ > P(X_j = -1) = p \quad P(X_j = 1) = q \quad P(X_j = 0) = 1 - p - q > $ > [!theorem] > > $ > \ev{(S(n)) = n(q - p)} \quad \var{S(n)} = n(p + q - (q - p)^2) > $ > *Proof*. > $ > \begin{align*} > \ev(X_j) &= 1 \cdot q + 0 \cdot (1 - p - q) - 1\cdot p = q - p \\ > \ev(S(n)) &= n\ev(X_j) = n(q - p) \\ > \var{X_j} &= \ev(X_j^2) - \ev(X_j)^2 \\ > \var{X_j} &= 1^2 \cdot q + 0^2 \cdot (1 - p - q) + (-1)^2 \cdot p - \ev(X_j)^2 \\ > \var{X_j} &= q + p - (q - p)^2 \\ > \var{S(n)} &= n(q + p - (q - p)^2) > \end{align*} > $