Let X and Y be random variables with μX = 1, σX = 2, μY = 3, σY = 1, and ρX,Y = 0.5. Find the means and variances of the following quantities.

a. X + Y

b. X - Y

c. 3X + 2Y

d. 5Y - 2X

Answer :

Step 1 of 1 :

Given,

let X and Y be independent random variables with = 1, =2, = 3, = 1 and

= 0.5

cov(X,Y) = () ( )

= (0.5) (2)(1)

= 1

The claim is to find the mean and variances for the following quantities

X + YX - Y3X + 2Y5Y - 2XStep 2 of 5 :

a)

The claim is to find the Mean and Variance for Quantity X + Y

Where, = +

Here, = 1 and = 3

Therefore, = 1 + 3

= 4

And = + + 2Cov(X,Y)

Where, = 2 and = 1

= 4 + 1 + 2(1)

= 7

For the Quantity X + Y mean is 4 and variance is 7.

Step 3 of 5 :

b)

The claim is to find the Mean and Variance for Quantity X - Y

Where, = -

Here, = 1 and = 3

Therefore, = 1 - 3

= -2

And = + - 2Cov(X,Y)

Where, = 2 and = 1

= 4 + 1 - 2(1)

= 3

For the Quantity X - Y mean is -2 and variance is 3.