Question

Let \( (X, Y)^T \) follow a bivariate normal distribution with
\[ E(X) = 3, \quad E(Y) = 4, \quad \text{Var}(X) = 25, \quad \text{Var}(Y) = 100, \quad \text{Cov}(X, Y) = 50 \rho, \]

• A. 0.08
• B. 0.8
• C. 0.32
• D. 0.5

Hint:

For conditional expectation in bivariate normal distributions, use the formula: \[ E(Y | X = x) = E(Y) + \frac{{Cov}(X, Y)}{{Var}(X)} (x - E(X)). \]

Answer 1

The Correct Option is A

For a bivariate normal distribution, the conditional expectation of \( Y \) given \( X = x \) is given by the formula: \[ E(Y | X = x) = E(Y) + \frac{{Cov}(X, Y)}{{Var}(X)} (x - E(X)). \] Substitute the given values: \[ E(Y | X = x) = 4 + \frac{50 \rho}{25} (x - 3). \] Given that \( E(Y | X = 5) = 4.32 \), we substitute \( X = 5 \) into the equation: \[ 4.32 = 4 + \frac{50 \rho}{25} (5 - 3). \] Simplifying the equation: \[ 4.32 = 4 + 4 \rho. \] Solving for \( \rho \): \[ 4.32 - 4 = 4 \rho \quad \Rightarrow \quad 0.32 = 4 \rho \quad \Rightarrow \quad \rho = \frac{0.32}{4} = 0.08. \] Thus, \( \rho = 0.08 \).