Discussion
The joint probability mass function PX,Y of the random variables X and Y is as given in the following table :
| PX,Y(x, y) | Y = 1 | Y = 2 | Y = 3 | Y = 4 | PX(x) |
|---|---|---|---|---|---|
| X = 1 | 0.02 | 0.04 | 0.06 | 0.08 | 0.20 |
| X = 2 | 0.03 | 0.06 | 0.09 | 0.12 | 0.30 |
| X = 3 | 0.05 | 0.10 | 0.15 | 0.20 | 0.50 |
| PY(y) | 0.10 | 0.20 | 0.30 | 0.40 |
Then, which of the following statements is true ?
Answer: C
X and Y are uncorrelated.
Every cell of the table equals the product of its marginals: \(0.02 = 0.20 \times 0.10\), \(0.06 = 0.30 \times 0.20\), \(0.15 = 0.50 \times 0.30\), and so on for all twelve cells. So X and Y are independent, and independent variables have zero covariance, hence are uncorrelated (and not positively correlated).
The means are \(E(X) = 0.2 + 0.6 + 1.5 = 2.3\) and \(E(Y) = 0.1 + 0.4 + 0.9 + 1.6 = 3.0\), so \(E(Y) \lt E(X)\) is false. And \(P(X \gt 2, Y \gt 2) = 0.15 + 0.20 = 0.35\), while \(1 - P(X \le 2, Y \le 3) = 1 - 0.30 = 0.70\), so the stated identity fails.
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