This is an exercise from an app with probability problems:
Let $X,Y$ be iid uniform random variables in $[0,1]$. Find the correlation coefficient $\rho$ of $X,Y$ conditional on $X+Y>1$.
Hint: Either use the definitions or (trick) regression of $X$ on $Y$.
I take the first hint and decide to follow the definitions, so I write $U:=X\mid X+Y>1$ and $V:=Y\mid X+Y>1$ and with this notation I want to calculate $$\rho(U,V)=\frac{Cov(U,V)}{\sqrt{Var(U)Var(V)}}=\frac{E[UV]-E[U]E[V]}{\sqrt{Var(U)Var(V)}}$$
I proceed by calculating the distribution function of $U$. For $0\le u\le 1$ \begin{align*}F_U(u)&=P(X\le u\mid X+Y>1)=\frac{P(X\le u, X+Y>1)}{P(X+Y>1)}=\frac{u(1-(1-u))\frac12}{\frac12}=u^2\end{align*} where, I calculated the probabilities geometrically (I have drawn a shape). Hence, $f_U(u)=2u$ for $0\le u\le 1$ and due to symmetry the same for $V$. This gives $$E[U]=\int_{0}^1u(2u)du=\frac23, \qquad E[U^2]=\int_{0}^1u^2(2u)du=\frac12$$ Hence, $Var(U)=\frac12-\left(\frac23\right)^2=\frac1{18}$ and due to symmetry $E[U]=E[V]$ and $Var(U)=Var(V)$. So, I may write $\rho$ as $$\rho(U,V)=\frac{E[UV]-E[U]^2}{Var(U)}=\frac{E[UV]-E[U]^2}{E[U^2]-E[U]^2}$$ It remains to calculate $E[UV]$. Again, from the drawing that I have, I find $$E[UV]=E[XY\mid X+Y>1]=\int_{0}^{1}\int_{1-x}^1xy\,dy\,dx=\frac5{24}$$ And now I substitute, but I get $\rho<-1$ which is certainly wrong $$\rho(U,V)=\frac{\frac5{24}-\left(\frac23\right)^2}{\frac12-\left(\frac23\right)^2}=-4.25$$ Can someone please tell where is my mistake? Can someone show me how to start with the second hint (regression)? Thanks!