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a random walk chooses a random bit independently and it moves left or right if the bit chosen is 0 or 1, respectively.

I was wondering if "choose a random bit independently" means that the random bit has a uniform distribution over {0,1}, and the walk sample this uniform distribution?

I remember that I often see expression such as "choose a random thing independently". Does it mean the random thing has a uniform distribution over its range? Does it have something to do with independence between events or random variables?

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    $\begingroup$ I think in this context it means each bit is independent of the previous ones. But this is not correctly formulated. $\endgroup$ – Elvis Dec 6 '12 at 15:56
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It's badly worded - perhaps the best interpretation you can put on it is that each choice is independent of previous choices. There's nothing to indicate that the distribution of the bit is uniform, except that if it weren't someone would quite likely say so.

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A bit can only have 1 of 2 values, 0 or 1, so it would not be a uniform distribution between 0 and 1 it means that either 0 or 1 is chosen and then the move is either left or right accordingly. The "independently" means that the probability of getting a 0 or 1 does not depend on the value of the previous random bit or the current location. Independent does not mean uniform (but it does not mean non-uniform either).

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    $\begingroup$ a "uniform distribution over {0,1}" means a discrete uniform distribution over the set containing just 0 & 1. So that's a 50% chance of each - a perfectly correct way to describe it. $\endgroup$ – Scortchi Dec 6 '12 at 20:22
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    $\begingroup$ In the next paragraph the question says "a uniform distribution over its range" which implied to me continuous uniform. But at the top he probably meant discrete as you point out. Either way independent does not mean (or contradict) uniform, either discrete or continuous. $\endgroup$ – Greg Snow Dec 6 '12 at 20:49

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