List of top Statistics Questions on Conditional Expectation

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

Let \( (X_1, X_2, X_3) \) follow the multinomial distribution with the number of trials being 100 and the probability vector \( \left( \frac{3}{10}, \frac{1}{10}, \frac{3}{5} \right) \).
Then \( E(X_2 | X_3 = 40) \) equals:

A fair die is rolled twice independently. Let \( X \) and \( Y \) denote the outcomes of the first and second roll, respectively. Then \[ E(X + Y \mid (X - Y)^2 = 1) \] The value of \( E(X + Y \mid (X - Y)^2 = 1) \) is _________ (round off to 2 decimal places).