# 2013-15 Bounded random variable

Let $$x, y$$ be real numbers satisfying $$y \geq x^2 + 1$$. Prove that there exists a bounded random variable $$Z$$ such that
$E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y.$
Here, $$E$$ denotes the expectation.

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# Solution: 2013-14 Nilpotent matrix

Let $$A, B$$ are $$N \times N$$ complex matrices satisfying $$rank(AB – BA) = 1$$. Prove that $$(AB – BA)^2 = 0$$.

The best solution was submitted by 김호진. Congratulations!

Similar solutions were submitted by 강동엽(+3), 김범수(+3), 김홍규(+3), 박민재(+3), 박지민(+3), 박훈민(+3), 안가람(+3), 어수강(+3), 엄문용(+3), 유찬진(+3), 이성회(+3), 이시우(+3), 이주호(+3), 장경석(+3), 전한솔(+3), 정동욱(+3), 정성진(+3), 정종헌(+3), 정우석(+3), 진우영(+3), Fardad Pouran(+3). Thank you for your participation.

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# Solution: 2013-13 Functional equation

Find all continuous functions $$f : \mathbb{R} \to \mathbb{R}$$ satisfying
$f(x) = f(x^2 + \frac{x}{3} + \frac{1}{9} )$
for all $$x \in \mathbb{R}$$.

The best solution was submitted by 강동엽. Congratulations!

Similar solutions were submitted by 김기현(+3), 김범수(+3), 김정섭(+3), 김호진(+3), 김홍규(+3), 박민재(+3), 박지민(+3), 박훈민(+3), 어수강(+3), 엄문용(+3), 윤성철(+3), 이명재(+3), 이성회(+3), 이시우(+3), 이주호(+3), 장경석(+3), 전한솔(+3), 정동욱(+3), 정성진(+3), 정종헌(+3), 조정휘(+3), 진우영(+3), 안가람(+2), 박경호(+2), 정우석(+2). Thank you for your participation.

Remark 1. As written in the rules, please submit the solution by 12PM on Wednesday. Any solution submitted after 12PM will not be graded.
Remark 2. Please write your name in the solution (not just in the email).

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# 2013-14 Nilpotent matrix

Let $$A, B$$ are $$N \times N$$ complex matrices satisfying $$rank(AB – BA) = 1$$. Prove that $$(AB – BA)^2 = 0$$.

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# Math Problem of the Week 2013 will resume at September 27.

Due to Thanksgiving (추석) break, POW will take a break next week as well. Next problem of the week will be posted at September 27.

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Find all continuous functions $$f : \mathbb{R} \to \mathbb{R}$$ satisfying
$f(x) = f(x^2 + \frac{x}{3} + \frac{1}{9} )$
for all $$x \in \mathbb{R}$$.