Random Walks

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The Lattice Random Walk is a statistical model, where at every step in time, a particle either goes right 1 unit, or left by 1 unit. Here, we show an example of the results of such a random walk after a certain number of timesteps, and provide a derivation.

Here, we derive the probability that a ball ends up in position $x$ after $t$ time-steps (notice that each y-level is a timestep). Let $x_i$ be the direction that the ball went on the $i$-th timestep, we have $x_i = \begin{cases} 1 & p = 0.5 \\ -1 & p = 0.5 \end{cases}$ $$X = \sum_{i=1}^t x_i$$ Notice that $x_i = 2*y_i -1 $ (where $y_i \sim \text{Bernoulli}(0.5)$), and so $$X = 2*Y - T$$ where $Y \sim \text{Binomial}(t,0.5)$