Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How to prove the following: $$ \lim_{n \to \infty} \int_0^1 \int_0^1 \cdots \int_0^1 \frac{x_1^2+x_2^2+ \cdots +x_n^2}{x_1+x_2+ \cdots +x_n} dx_1 dx_2 \cdots dx_n = \frac23 $$

I would really appreciate if you could help me!

share|improve this question

1 Answer 1

up vote 10 down vote accepted

Let $X_1, X_2,\dots$ be independent, uniform$(0,1)$ random variables. By the law of large numbers we have $$\begin{eqnarray*} {X_1+\cdots + X_n\over n}&\to& \mathbb{E}(X)={1\over 2}\\ {X_1^2+\cdots + X^2_n\over n}&\to& \mathbb{E}(X^2)={1\over 3}\\ \end{eqnarray*} $$ in probability as $n\to\infty$. Therefore $${X_1^2+\cdots +X^2_n\over X_1+\cdots +X_n}={X_1^2+\cdots +X^2_n\over n}\cdot{n\over X_1+\cdots +X_n}\to {2\over 3}$$ in probability as $n\to \infty$. The ratio random variables ${X_1^2+\cdots +X^2_n\over X_1+\cdots +X_n}$ are bounded below by zero and above by one. This guarantees convergence of the expectations, as well. So $$\mathbb{E}\left({X_1^2+\cdots +X^2_n\over X_1+\cdots +X_n}\right)\to{2\over 3}$$ which is the required result.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.