SOLUTION: Let X and Y be jointly normal random variables with parameters µ_X = 1, (σ_X)^2= 1, µ_Y = 0, (σ_Y)^2= 4, and ρ = 1/2. (a) Find P(2X + Y

SOLUTION: Let X and Y be jointly normal random variables with parameters µ_X = 1, (σ_X)^2= 1, µ_Y = 0, (σ_Y)^2= 4, and ρ = 1/2. (a) Find P(2X + Y < 3). (b) Find P(Y > 1|X = 2). (c)

Question 1204126: Let X and Y be jointly normal random variables with parameters µ_X = 1, (σ_X)^2= 1, µ_Y = 0, (σ_Y)^2= 4, and ρ = 1/2.
(a) Find P(2X + Y < 3).
(b) Find P(Y > 1|X = 2).
(c) Find conditional expectation of Y given X = 2.

Answer by ElectricPavlov(122) (Show Source): You can put this solution on YOUR website!
**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!

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