Question: A manufacturer of television sets sells two principal models. Define \(X\) next December. The marketing staff estimates that the joint probabilities \({{P}_{XY}}(x,y)\) are:

1. Find \(P(X=1, Y=2)\).
2. Find \(P(X \leq 2, Y \leq 2)\).
3. Find \(P_{x}(x)\) and \(P_{y}(y)\).
4. Are \(X\) and \(Y\) independent?
5. Calculate \(\mu_{x}\) and \(\mu_{Y}\).
6. Calculate \(\sigma_{x}\) and \(\sigma_{Y}\).
7. Calculate \(\operatorname{Cov}(X, Y)\).
8. Calculate \(\rho_{X Y}\).
9. Find \(P_{Y X}(y \mid X=2)\).
10. Calculate \(E(Y \mid X=2)\).

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