Curiosities · Foundation · 5 min
Random walks
A random path can wander even when its expected displacement is zero.
WHAT HAPPENS IF…For n=20 and p=0.75, what are mean and variance?
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Each independent ±1 step has mean 2p−1 and variance 4p(1−p). Means and independent variances add over n steps. For a fair walk, expected displacement is zero but standard deviation is √n. Resetting the same seed and controls reproduces the same sample.
A detail that matters
One trajectory need not stay near its expectation.
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