Suppose $X_1,\ldots,X_n$ are drawn i.i.d from a uniform distribution on $[0,1]$ and let $x$ be the random vector $(X_1,\ldots,X_n)$. Then consider the random variable $Y_v = v^\top x $ for all $v \in \{0,1\}^n$ — note there are $2^n$ such random variables $Y_v$. Given a range (say $\left[\frac{1}{2}n, \frac{2}{3}n\right]$ for example), what is the expected number of random variables $Y_v$ that take a value in this range; i.e. what is $$\sum_{v \in \{0,1\}^n}{\Large{𝟙}}_{Y_v \in \left[\frac{1}{2}n, \frac{2}{3}n\right]}?$$
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1$\begingroup$ Cross-posted on Mathematics SE. $\endgroup$– TheSimpliFire ♦Oct 30, 2019 at 19:25
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$\begingroup$ Hint: the expected value of $Y_{v}$ is $\frac{n}{4}$. $\endgroup$– MaartenOct 30, 2019 at 22:02
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2$\begingroup$ I'm voting to close this question as off-topic because it's already crossposted to Mathematics and is far more relevant there. $\endgroup$– Geoffrey BrentOct 30, 2019 at 23:36
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