[Math] A question about random walk in 1 dimension

probabilityprobability theoryrandom walk

For a simple random walk problem in 1 D, the expected position of the particle in $n$ step is $E(X_n)=n(p-q)$ so the distance from origin should be $=E(X_n)$ but according to Mean distance from origin after $N$ equal steps of Random-Walk in a $d$-dimensional space. the expected distance from the origin is different. So why is it?

Best Answer

Position can be negative, distance is the absolute value of position.
So the expected distance is the expected value of the absolute value of the position, not the absolute value of the expected position.

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