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I am an undergraduate student studying some elementary calculus and statistics. In my honor calculus class, my professor gave one of final exam problem:

$$\lim_{n \to \infty} \int_{[0,1]^{n}} \left(\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)^{2013.1214} d \mathbf{x} = ?$$

At first sight I've brutally tried to calculate it directly: changing variables with $x_{1}=y_{1}$, $x_{1}+x_{2}=y_{2}$, $\cdots$, $x_{1}+\cdots+x_{n}=y_{n}$. Of course it has failed and the time was ran out.

After the exam, I asked this problem to my friend (who did not took this exam). Just after 10 second he saw this problem, he said: "each $x_{i}$ is picked from $[0,1]$ independently, so as $n \to \infty$, its average will go to $\frac{1}{2}$, so the value of limit would be $(\frac{1}{2})^{2013.1214}$. Try to justify this by using the central limit theorem"

I was really shocked, that just after first glance he got how to solve this and realized the meaning of this problem, and felt I became an almost fool and complete idiot, becuase even though I already took the elementary statistics class to learn the central limit theorem, when the time to apply to solve problem was came, I couldn't think to apply that for this calculus problem - but my friend, just after the first glimpse, realized that the essential meaning of this problem is asking where the certain power expectation of average of uniform distribution goes as the size of sample gets bigger.

How can I raise my intuition to catch the essential meaning contained in the mathematical problem and move on from that? Just think over and over again? Or are these kinds of intuition just a gift from the god?

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    $\begingroup$ A big part of the answer is: Accept that this will happen sometimes and keep at it. $\endgroup$ – Michael Hardy Dec 16 '13 at 3:51
  • $\begingroup$ The theological part of the question is probably outside of the subject matter of this forum. The other part I think might fit if there are good answers. $\endgroup$ – Michael Hardy Dec 16 '13 at 3:53
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    $\begingroup$ Practice. Read books on problem solving. That's all you can do, really. $\endgroup$ – Potato Dec 16 '13 at 4:39
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    $\begingroup$ You develop the intuition by braking it. You have to search constantly for the statements which contradict your intuition and develop better intuition this way. I hope it happened with this problem, so you are on the right track. This is bit more specific than just say "practice"; hope it helps :) $\endgroup$ – Anton Petrunin Dec 16 '13 at 5:13
  • $\begingroup$ The answer isn't quite as simple as described unless explicitly stating something to do with Monte Carlo Integration for me i.e. using LLN twice. E.g. $\lim_{n \to \infty} \int_{[0,2]^{n}} \left(\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)^{2013.1214} d \mathbf{x}$ is a little different. $\endgroup$ – rwolst Dec 16 '13 at 15:55
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I don't know. Was it my ‘feminine intuition’ that helped me realize that $\displaystyle\sum_{n=0}^\infty\left[\frac{(2n-3)!!}{(2n)!!}\right]^2=\frac4\pi$, or merely plucking $n=\frac12$ into Vandermonde's formula $\displaystyle\sum_{k=0}^n{n\choose k}^2={2n\choose n}$, and using the fact that $\Gamma\left(\frac12\right)=\sqrt\pi$ ? :-)

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