What is the expected number of bit $1$'s when writing a random integer from $1$ to $1024$ in binary?
I noticed that $1024 = 2^{10}$, so maybe linearity of expectation could help here?
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Why expected? There is no random involved, the answer is $1024\cdot 5+1$ – Hagen von Eitzen Sep 21 '14 at 04:05
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Should your question be "If I pick a random integer from $1$ to $1024$, how many bit $1's$ are expected?" – Semiclassical Sep 21 '14 at 04:05
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Yes, @Semiclassical – William Yi Sep 21 '14 at 04:06
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Ok. I've modified your question accordingly. In the future, please modify your question if you realize it isn't sufficiently clear. – Semiclassical Sep 21 '14 at 04:09
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If the binary number were selected uniformly from the range $\{000000000_2, 111111111_2\}$, the number of 1's would have a binomial distribution ${\cal Bin}(9, \tfrac 1 2)$, and an expected number of $4.5$.
Can you adjust this for a selection from the range of $\{0000000001_2, 1000000000_2\}$?
Graham Kemp
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