Q:

Calculate correlation coefficient for discrete random variable

From a population consisting of the numbers: $\lbrace 1,2 \ldots 10 \rbrace$, two samples are chosen from it without replacement. If the random variable denoting the first choice is X and the second choice is $Y$, what is the correlation coefficient ($\rho$) between $X$ and $Y$

A:

Assuming $X$ is uniform on $\{1, 2, \dots, 10\}$ we have 
$$
\mathbb E X = \frac{1 + 2 + \dots + 10}{10} = 5.5, \quad \mathbb{V}ar X = \frac{1^2 + \dots + 10^2}{10} - \mathbb E X^2 = 8.25.
$$
To compute the expected value of $Y$ write 
$$
\mathbb E Y = \sum_{x \in [10]} \mathbb E[Y ~|~ X=x] \mathbb{P}(X = x),
$$
where $[n] := \{1, 2, \dots, n \}.$ Let us first compute $\mathbb E Y:$
$$
\mathbb E Y = \frac 1{10}\left[\frac {1 + \dots + 9}{9} + \dots + \frac{2 + \dots + 10}{9}\right] = \frac{55}{10} = 5.5.
$$
$$
\mathbb Var Y = \frac 1{10}\left[\frac {1^2 + \dots + 9^2}{9} + \dots + \frac{2^2 + \dots + 10^2}{9}\right] - \mathbb E Y^2 = 8.25
$$
For $\mathbb E XY$ we have 
$$
\mathbb E XY = \frac{1}{10}\left[\frac{1 + 2 + \dots + 10}{9\cdot 10} \cdot \sum_{x, y \in [10] \setminus x} y \right] = 30.25.
$$
Hence, 
$$
\rho(X, Y) = \frac{\mathbb Cov(X, Y)}{\sqrt{\mathbb Var X \mathbb Var Y}} = \frac{30.25 - 5.5^2}{8.25} = 0. 
$$
As for me, it is counterintuitive that the answer is $0$ and I suspect that there must be much simpler solution. 

