Solution
Correlation of a Number with Its Square
Show the problem again
Let $X$ be uniform on $[0,1]$. What is the correlation between $X$ and $X^2$? Give an exact closed form (a decimal rounded to 3 or 4 places is also accepted).
Worked solution
Use $E[X^k] = 1/(k+1)$. Then $\operatorname{Cov}(X, X^2) = 1/4 - (1/2)(1/3) = 1/12$, $\operatorname{Var}X = 1/12$ and $\operatorname{Var}X^2 = 1/5 - 1/9 = 4/45$. The correlation is $\dfrac{1/12}{\sqrt{(1/12)(4/45)}} = \sqrt{\dfrac{45}{48}} = \dfrac{\sqrt{15}}{4} \approx 0.968$. A perfectly deterministic monotone curve is still not perfectly correlated, because correlation measures only linear association.
Source: Standard calculation in correlation texts; no single originator. Statement written for AxiomIQ.