Probability and Random variable covariance

Question

Suppose a box has 3 balls labeled 1, 2, and 3. Two balls are selected without replacement from the box. Let $X$ be the number on the first ball and let $Y$ be the number on the second ball. Compute $\mathrm{Cov}(X; Y )$ and $\rho (X; Y )$.

Answer 1

Call up the usual suspects: the definitions.$$\begin{split}\mathsf {Var}(X)&=\mathsf E(X^2)-\mathsf E(X)^2\\\mathsf {Var}(Y)&=\mathsf {Var}(X)\\\mathsf {Cov}(X,Y)&=\mathsf E(XY)-\mathsf E(X)\mathsf E(Y)\\\rho_{\tiny X,Y} &= \dfrac{\mathsf{Cov}(X,Y)}{\sqrt{\mathsf {Var}(X)}\;\sqrt{\mathsf {Var}(Y)}}\\[6ex]\mathsf E(X)&=\sum_{x=1}^3x\;\mathsf P(X=x)\\\mathsf E(X^2)&=\sum_{x=1}^3x^2\;\mathsf P(X=x)\\\mathsf E(XY)&=\underset{y\neq x}{\sum_{x=1}^3\sum_{y=1}^3} xy\;\mathsf P(X=x,Y=y)\end{split}$$