$Prove that $\lim_{n\rightarrow \infty} \int_{[0,1]^n}\frac{|x|}{\sqrt{n}}=\frac{1}{\sqrt{3}}$$

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$Prove that $\lim_{n\rightarrow \infty} \int_{[0,1]^n}\frac{|x|}{\sqrt{n}}=\frac{1}{\sqrt{3}}$$

Prove that $$\lim_{n\rightarrow \infty} \int_{[0,1]^n}\frac{|x|}{\sqrt{n}}=\frac{1}{\sqrt{3}},$$
where $[0,1]^n=[0,1]\times \dots \times [0,1]$ is the unit cube in $\mathbb{R}^n$.

Calculus Probability Statistics

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Let $𝑋1,𝑋2,... $ be an i.i.d. sequence of random variables with uniform distribution on $[0,1]$, then the given limit is

$$lim \frac{1}{\sqrt{n}}𝐸(𝑋_1^2+𝑋_2^2+⋯+𝑋_𝑛^2)^{1/2}.$$

By the Strong Law of Large Numbers we have
\[\frac{E(𝑋_1^2+𝑋_2^2+⋯+𝑋_𝑛^2)}{n}→\int_0^1 𝑥^2𝑑𝑥=\frac{1}{3}\]
and the result follows from the above.

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$Prove that $\lim_{n\rightarrow \infty} \int_{[0,1]^n}\frac{|x|}{\sqrt{n}}=\frac{1}{\sqrt{3}}$$

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