Discussion
Let \(X_{1} \sim N(0, 1)\) and
\[ X_{2} = \begin{cases} -X_{1} & -3 \le X_{1} \le 3 \\ X_{1} & \text{otherwise} \end{cases} \]
Then which of the following statements is correct ?
Answer: A
\(X_{1}\) and \(X_{2}\) are negatively correlated.
\(X_{2}\) is again N(0, 1), by the symmetry of the normal distribution, and \(X_{1}X_{2} = -X_{1}^{2}\) on \(|X_{1}| \le 3\) and \(+X_{1}^{2}\) outside. So
\[ \begin{aligned} \mathrm{Cov} &= -E[X_{1}^{2}; |X_{1}| \le 3] \\ &\quad + E[X_{1}^{2}; |X_{1}| \gt 3] \end{aligned} \]
Almost all of \(E[X_{1}^{2}] = 1\) comes from \(|X_{1}| \le 3\) (the tail beyond 3 contributes about 0.03), so the covariance is about \(-0.97 + 0.03 \lt 0\).
The correlation is not \(-1\), because \(X_{2} = X_{1}\) on the tails, so they are not perfectly correlated. They are not jointly normal either: \(X_{1} + X_{2}\) equals 0 with probability about 0.997 but is not identically 0, which is impossible for a normal variable.
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