Question

Let Y1 and Y2 be independent random variables with Y1 ∼ N(1,3)

and Y2 ∼ N(2,5). If W1 = Y1 +2Y2 and W2 = 4Y1 −Y2, what is the
joint distribution of W1 and W2?

Answer 1

To find the joint distribution of W1 and W2, calculate the mean and variance of W1 and W2 based on the distributions of Y1 and Y2. Since Y1 and Y2 are independent, the joint distribution of W1 and W2 is a bivariate normal distribution.

Step-by-step explanation:

To find the joint distribution of W1 and W2, we need to consider the distributions of Y1 and Y2 and then use the properties of linear combinations of random variables.

Since Y1 ∼ N(1,3) and Y2 ∼ N(2,5), we can calculate the mean and variance of W1 and W2:

Mean of W1 = 1 + 2(2) = 5

Variance of W1 = (2)^2 * 5 = 20

Mean of W2 = 4(1) - 2 = 2

Variance of W2 = (4)^2 * 3 + (-1)^2 * 5 = 63

Since Y1 and Y2 are independent, the joint distribution of W1 and W2 is a bivariate normal distribution.

Answer 2

The joint distribution of W1 and W2 can be expressed as f(w1,w2) = f(y1)f(y2) / |J|, where f(y1) and f(y2) are the probability density functions of Y1 and Y2.

Step-by-step explanation:

To find the joint distribution of W1 and W2, we need to find the mean and variance of W1 and W2. Using the properties of the normal distribution, we can calculate:

E(W1) = E(Y1 + 2Y2) = E(Y1) + 2E(Y2) = 1 + 2(2) = 5

Var(W1) = Var(Y1 + 2Y2) = Var(Y1) + 4Var(Y2) = 3 + 4(5) = 23

E(W2) = E(4Y1 - Y2) = 4E(Y1) - E(Y2) = 4(1) - 2 = 2

Var(W2) = Var(4Y1 - Y2) = 16Var(Y1) + Var(Y2) = 16(3) + 5 = 53

The joint distribution of W1 and W2 is given by:

f(w1,w2) = P(W1 = w1, W2 = w2)

Using the properties of independent normal random variables, we can express the joint distribution as follows:

f(w1,w2) = f(y1,y2) / |J|

where f(y1,y2) is the joint probability density function of Y1 and Y2, and |J| is the determinant of the Jacobian matrix. Since Y1 and Y2 are independent, f(y1,y2) = f(y1)f(y2), where f(y1) and f(y2) are the probability density functions of Y1 and Y2, respectively.

Therefore, the joint distribution of W1 and W2 can be expressed as:

f(w1,w2) = f(y1)f(y2) / |J|