Vtyjkyuk

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 $Y$

Assuming $X$ is uniform on $1, 2, dots, 10$ we have
$$
mathbb E X = frac1 + 2 + dots + 1010 = 5.5, quad mathbbVar X = frac1^2 + dots + 10^210 - mathbb E X^2 = 8.25.
$$
To compute the expected value of $Y$ write
$$
mathbb E Y = sum_x in [10] mathbb E[Y ~|~ X=x] mathbbP(X = x),
$$
where $[n] := 1, 2, dots, n .$ Let us first compute $mathbb E Y:$
$$
mathbb E Y = frac 110left[frac 1 + dots + 99 + dots + frac2 + dots + 109right] = frac5510 = 5.5.
$$
$$
mathbb Var Y = frac 110left[frac 1^2 + dots + 9^29 + dots + frac2^2 + dots + 10^29right] - mathbb E Y^2 = 8.25
$$
For $mathbb E XY$ we have
$$
mathbb E XY = frac110left[frac1 + 2 + dots + 109cdot 10 cdot sum_x, y in [10] setminus x y right] = 30.25.
$$
Hence,
$$
rho(X, Y) = fracmathbb Cov(X, Y)sqrtmathbb Var X mathbb Var Y = frac30.25 - 5.5^28.25 = 0.
$$

As for me, it is counterintuitive that the answer is $0$ and I suspect that there must be much simpler solution.