Transition Probabilities in a Random Walk

Topics: Random Walk

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(theorem)

Let $\left\{X_n\mid n\in \mathbb{N}\right\}$ be a random walk.

For any two integers $x$ and $k$, either both even or both odd, such that $-n\le x\le n$, we have:

• …when $X_0=0$ (it’s a simple random walk):

$$\mathbb{P}\left(X_n=x\mid X_0=0\right)=\left(\genfrac{}{}{0ex}{}{n}{\frac{1}{2}(n+x)}\right)p^{\frac{1}{2}(n+x)}{q}^{\frac{1}{2}(n-x)}$$

• …when $X_0=y\ne 0$, then:

$$\mathbb{P}\left(X_n=x\mid X_0=y\right)=\left(\genfrac{}{}{0ex}{}{n}{\frac{1}{2}(n+x-y)}\right)p^{\frac{1}{2}(n+x-y)}{q}^{\frac{1}{2}(n-x+y)}$$