Question

Let \( (X, Y) \) have a bivariate normal distribution with the joint probability density function \[ f_{X,Y}(x,y) = \frac{1}{\pi} e^{\left( \frac{3}{2} xy - \frac{25}{32} x^2 - 2 y^2 \right)} \] Then \[ E(XY) = _________ \text{ (round off to 2 decimal places).} \]

Hint:

In bivariate normal distributions, if \( X \) and \( Y \) are independent, the covariance \( E(XY) \) is zero.

Answer 1

Correct Answer: 3

For a bivariate normal distribution, the expectation \( E(XY) \) is given by the covariance between \( X \) and \( Y \), which is zero in the case of independent variables. Therefore, we calculate the covariance from the given joint distribution, which simplifies to: \[ E(XY) = 3. \] Thus, the value is \( 3 \).