Question

Let \((X,Y)\) be a random vector having the bivariate normal distribution with \(E(X)=2\), \(E(Y)=10\), \(Var(X)=9\), \(Var(Y)=25\), and the correlation coefficient between \(X\) and \(Y\) equals \(\rho>0\). If

Hint:

For bivariate normal distributions, the conditional variance is reduced by the factor \((1-\rho^2)\).

Answer 1

Correct Answer: 0.6

Step 1: Conditional law of $Y$ given $X=2$.
At $X=2=\mu_X$ the mean stays $\mu_Y=10$, and the variance shrinks to $25(1-\rho^2)$.

Step 2: Standardise the probability.
$P(4<Y<16|X=2)=0.8664$ becomes $P\left(|Z|<\frac6{5\sqrt{1-\rho^2}}\right)=0.8664$.

Step 3: Read the $z$ value.
This gives $\Phi\left(\frac6{5\sqrt{1-\rho^2}}\right)=0.9332=\Phi(1.5)$, so $\frac6{5\sqrt{1-\rho^2}}=1.5$.

Step 4: Solve for $\rho$.
Then $\sqrt{1-\rho^2}=\frac6{7.5}=0.8$, so $1-\rho^2=0.64$, $\rho^2=0.36$, and with $\rho>0$, $\rho=0.6$.

Step 5: Conclude.
\[ \boxed{0.60} \]