First of all, I'm apologizing for such a basic question.
I've just started with probability & statistics and I came across this example:
Let $X, Y$ be random indpendent variables defined with $$ \begin{pmatrix} 0 & 1 \\ \frac{1}{2} &\frac{1}{2} \end{pmatrix} $$ Let $U=\min\left \{ X,Y\right \}$ , $V=\max\left \{ X,Y\right \}$. Find the correlation coefficient $\rho_{U,V}$
I know how to calculate $\rho_{U,V}$ ,but what I don't know is how to determine $U$ and $V$?
$X$ and $Y$ are iid Bernoulli$\left(\frac{1}{2}\right)$ i.e. joint probability mass function of the two random variables is $p_{X,Y}(a, b) = \frac{1}{4}$ for $a\in\{0,1\}$ and $b\in\{0,1\}$.
Joint distribution of $U = \min\{X, Y\}$ and $V = \max\{X, Y\}$ is therefore $p_{U,V}(0, 0) = \Pr(U = 0, V = 0) = \Pr(X = 0, Y = 0) = \frac{1}{4}$ $p_{U,V}(0, 1) = \Pr(U = 0, V = 1) = \Pr(X = 0, Y = 1) + \Pr(X = 1, Y = 0)= \frac{1}{2}$ $p_{U,V}(1, 1) = \Pr(U = 1, V = 1) = \Pr(X = 1, Y = 1) = \frac{1}{4}$
Marginal distribution of $V$ is $p_V(0) =\frac{1}{4}$ and $p_V(1) =\frac{3}{4}$ In other words, $V$ is Bernoulli$\left(\frac{3}{4}\right)$
and marginal distribution of $U$ is $p_U(0) =\frac{3}{4}$ and $p_U(1) =\frac{1}{4}$ i.e., $U$ is Bernoulli$\left(\frac{1}{4}\right)$
Since $UV = U$, distribution of the product $UV$ is Bernoulli$\left(\frac{1}{4}\right)$.
To calculate the correlation between $U$ and $V$, we will find the Covariance between $U$ and $V$ and variance of $U$ and variance of $V$.
$Cov(U, V) = \mathbb{E}(UV) - \mathbb{E}(U)\mathbb{E}(V) = \frac{1}{4} - \left(\frac{1}{4}\times\frac{3}{4}\right) =\frac{1}{16} $
$\mathbb{V}(U) = \mathbb{E}(U^2) - (\mathbb{E}(U))^2 = \frac{1}{4} - \left(\frac{1}{4}\right)^2 =\frac{3}{16} $
$\mathbb{V}(V) = \mathbb{E}(V^2) - (\mathbb{E}(V))^2 = \frac{3}{4} - \left(\frac{3}{4}\right)^2 =\frac{3}{16} $.
Therefore, $\displaystyle\rho_{U,V} = \frac{Cov(U, V)}{\sqrt{\mathbb{V}(U)\mathbb{V}(V)}} = \frac{1}{3}$