# What's the expected number of subsets of iid random variables with sum in given range? [closed]

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 $$\sum_{v \in \{0,1\}^n}{\Large{𝟙}}_{Y_v \in \left[\frac{1}{2}n, \frac{2}{3}n\right]}?$$

• Cross-posted on Mathematics SE. – TheSimpliFire Oct 30 '19 at 19:25
• Hint: the expected value of $Y_{v}$ is $\frac{n}{4}$. – Maarten Oct 30 '19 at 22:02
• I'm voting to close this question as off-topic because it's already crossposted to Mathematics and is far more relevant there. – Geoffrey Brent Oct 30 '19 at 23:36