Question
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
Wequals5Xminus6Y
Answers
GPT 3.5
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.
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.