Question 1204126
**a) Find P(2X + Y < 3)**

* **Define a new random variable:** Let Z = 2X + Y 

* **Find the mean and variance of Z:**

    * E[Z] = E[2X + Y] = 2E[X] + E[Y] = 2(1) + 0 = 2
    * Var(Z) = Var(2X + Y) = 4Var(X) + Var(Y) + 2 * 2 * Cov(X, Y) 
      * Cov(X, Y) = ρ * σ_X * σ_Y = (1/2) * 1 * 2 = 1
    * Var(Z) = 4 * 1 + 4 + 2 * 1 = 10

* **Determine the distribution of Z:**

    * Since X and Y are jointly normal, any linear combination of them (like Z) is also normally distributed.

* **Standardize Z:**

    * Let W = (Z - E[Z]) / sqrt(Var(Z)) 
      * W = (Z - 2) / sqrt(10) 
      * W follows a standard normal distribution (N(0, 1))

* **Calculate the probability:**

    * P(2X + Y < 3) = P(Z < 3) 
      * P(Z < 3) = P( (Z - 2) / sqrt(10) < (3 - 2) / sqrt(10) ) 
      * P(Z < 3) = P(W < 1 / sqrt(10)) 
      * Use a standard normal distribution table or software to find P(W < 1 / sqrt(10)) 

**b) Find P(Y > 1 | X = 2)**

* **Conditional Distribution of Y given X:**

    * When X and Y are jointly normal, the conditional distribution of Y given X is also normal.
    * The conditional mean of Y given X is:
        * E[Y | X] = μ_Y + ρ * (σ_Y / σ_X) * (X - μ_X) 
          * E[Y | X = 2] = 0 + (1/2) * (2 / 1) * (2 - 1) = 1

    * The conditional variance of Y given X is:
        * Var(Y | X) = σ_Y² * (1 - ρ²) 
          * Var(Y | X) = 4 * (1 - (1/2)²) = 4 * (3/4) = 3

* **Calculate the probability:**

    * P(Y > 1 | X = 2) = P( (Y - E[Y | X = 2]) / sqrt(Var(Y | X)) > (1 - 1) / sqrt(3) ) 
      * P(Y > 1 | X = 2) = P(Z > 0) 
        * where Z is a standard normal random variable. 
      * P(Y > 1 | X = 2) = 0.5 

**c) Find the conditional expectation of Y given X = 2**

* As calculated in part (b):
    * E[Y | X = 2] = 1

**In summary:**

* **(a) P(2X + Y < 3)** requires standardizing the linear combination of X and Y and then using a standard normal distribution table.
* **(b) P(Y > 1 | X = 2)** utilizes the properties of the conditional distribution of Y given X in a jointly normal distribution.
* **(c) The conditional expectation of Y given X = 2** is calculated directly using the formula for the conditional mean.

I hope this comprehensive explanation is helpful!