The random vector X = (X1, X2, X3, X4)T has a mean vector and dispersion matrix ,µ =23−23and covariance matrix Σ =3 1 0 41 5 0 40 0 2 54 4 5 8,LetY1 =X2 + X3 − 2X4 + 6Y2 =X1 − 2X3 + X4Calculate the correlation coefficient of (Y1, Y2)5. Let X = (X1, X2) be a random vector where X1 and X2 are independent and X1 ∼Exp15and X2 ∼ Exp 110,STAT 332 Exercise 1: EO/ES-Y/KDA Page 3 of 7

I'm sorry, but the question you've asked seems to be incomplete and the mean vector and covariance matrix are not properly formatted, which makes it difficult to provide an accurate answer. However, I can explain how to calculate the correlation coefficient of two variables in general.

The correlation coefficient (r) of two variables Y1 and Y2 can be calculated using the following formula:

r = Cov(Y1, Y2) / (std_dev(Y1) * std_dev(Y2))

Where:

• Cov(Y1, Y2) is the covariance of Y1 and Y2
• std_dev(Y1) is the standard deviation of Y1
• std_dev(Y2) is the standard deviation of Y2

The covariance can be calculated using the formula:

Cov(Y1, Y2) = E[(Y1 - E[Y1]) * (Y2 - E[Y2])]

Where E denotes the expected value.

The standard deviation can be calculated using the formula:

std_dev(Y) = sqrt(E[(Y - E[Y])^2])

Once you have calculated the covariance and standard deviations, you can substitute these values into the formula for r to find the correlation coefficient.

Please provide the correct and complete details so that I can help you solve it accurately.