A random variable X is normally distributed with a mean of 100 and a variance of 100, and a random variable Y is normally distributed with a mean of 160 and a variance of 256. The random variables have a correlation coefficient equal to negative 0.5. Find the mean and variance of the random variable below.
Wequals5Xminus6Y
To find the mean and variance of the random variable W = 5X - 6Y, we can use the properties of expected value and variance.
First, let's find the expected value (mean) of W:
E(W) = E(5X - 6Y)
E(W) = 5E(X) - 6E(Y)
E(W) = 5(100) - 6(160)
E(W) = 500 - 960
E(W) = -460
So, the mean of the random variable W is -460.
Next, let's find the variance of W:
Var(W) = Var(5X - 6Y)
Var(W) = 5^2Var(X) + (-6)^2Var(Y) - 2(5)(-6)Cov(X,Y)
Var(W) = 5^2(100) + (-6)^2(256) - 2(5)(-6)(-0.5)(√(100)(√(256))
Var(W) = 2500 + 1296 + 120
Var(W) = 3916
Therefore, the variance of the random variable W is 3916.