BM – B(3)^B(5) independent@@

Jargon: B3 or B(3) means the random position at time 3. I think this is a N@T.

Q: Are the 2 random variables B3 and B5 independent?
A: no. Intuitively, when B3 is very high, like 3 sigma above the mean (perhaps a value of 892), then B5 is likely to Remain high, because for the next 2 seconds, the walker follows a centered, symmetric random walk, centered at the realized value of 892.

We know the increment from time 3 to 5 is ind of all previous values. Let d be that increment.

B5 = B3 + d, the sum of two ind Normal RV. It’s another normal RV, but dependent on the two!

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