Discussion
Suppose X has bivariate normal distribution with mean vector (3, 4)' and \(V(X_{1}) = 2\). Further, it is known that \(X_{1}\) and \(X_{2}\) are positively correlated. Which of the following can be the conditional distribution of \((X_{1} \mid X_{2} = 2)\) ?
Answer: B
The possible conditional distribution is N(2.6, 1.84).
For a bivariate normal, \(E(X_{1} \mid X_{2} = x)\) \(= \mu_{1} + \rho\frac{\sigma_{1}}{\sigma_{2}}(x - \mu_{2})\) and \(V(X_{1} \mid X_{2}) = \sigma_{1}^{2}(1 - \rho^{2})\). Here \(\mu_{1} = 3\), \(\mu_{2} = 4\), \(\sigma_{1}^{2} = 2\) and \(\rho \gt 0\). Since \(x - \mu_{2} = 2 - 4 = -2 \lt 0\), the conditional mean is below 3, and the conditional variance is below 2.
Only N(2.6, 1.84) satisfies both: mean less than 3 and variance less than 2. N(1.5, 2.64) and N(3.5, 2.84) have variance above 2, and N(3.5, 1.52) has a mean above 3, which would need negative correlation.
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