So in one example they state the following:
(1): If $X\sim \text{unif[0,1]}$ and given $X=x$ and $Y\sim\text{unif}[0,x],$ what is $\rho(X,Y)?$
From previous treatment of this example we know that the means are $E[X]=1/2$ and $E[Y]=1/4$...
Then they proceed solving the problem. However, on the next page they have the following example
(2): Let $X\sim\text{unif}[0,1]$ and given $X=x$, let $Y\sim\text{unif}[0,x^2].$ Find $E[Y|X=x]$ and $l(x).$
It's clear from the description that $E[Y|X=x]=x^2/2....$
Wait...what? Can someone explain to me how they computed $E[Y]$ in example (1)? Shouldn't it say $E[Y|X=x]$ there too?
In (1) they have integrated the answer and in (2) they just divide the range of $Y$ by $2$. I don't see the difference here. Need some light sheded on this.