In order to measure precisely the influence of one variable $Y$ over $X$ I wanted to use the mutual information because so far I believed that correlation coefficient (Pearson) was only limited to linear relation.

But I realized that even in non-linear cases it gives good results.

**Example in Matlab:**

$X$ is a uniform random variable and $I$ is the range of $X$ 

$Y=X^{2},I=[0,100]: \rho=0.96$

$Y=\sqrt{X} ,I=[0,100]: \rho=0.94$

$Y=exp(X),I=[0,10]: \rho=0.70$

$Y=log(X),I=[0,100]: \rho=0.84$

Why the correlation is quite high in all those non-linear cases? I am beginning to think that correlation is only weak when the monotony of $X$ and $Y$ is not the same. Otherwise it is not so bad.

Do some people have a point of view on this?