1. Assume that q and z are two random variables that are perfectly positively correlated. q takes the value of 20 with probability 0.5 and the value of zero with probability 0.5, while z takes the value of 10 with probability 0.5 and the value of zero with probability 0.5. What is the COVARIANCE of q and z?
(A) 50.
(B) 100.
(C) 0.
(D) 1.
(E) There is not enough information to tell.
(Solution is B, 100)
Because they are perfectly postively correlated, the covariance between the q and z is 1.00
Because, (while not explained or pointed out), the possible answers are expressed in percentage terms and the decimal point is moved over 2 places, the answer you seek is B.
To find the covariance between two random variables, you need to know their joint probability distribution and their respective means.
In this case, you have the joint probability distribution:
P(q=20, z=10) = 0.5
P(q=20, z=0) = 0
P(q=0, z=10) = 0
P(q=0, z=0) = 0.5
We also need to determine the means of the variables q and z. The mean of a random variable is the weighted average of its values, where the weights are the corresponding probabilities.
Mean(q) = (20 x 0.5) + (0 x 0.5) = 10
Mean(z) = (10 x 0.5) + (0 x 0.5) = 5
Now, we can calculate the covariance using the formula:
Cov(q, z) = E[(q - mean(q))(z - mean(z))]
Cov(q, z) = [20 - 10][10 - 5] x 0.5 + [0 - 10][10 - 5] x 0.5 + [20 - 10][0 - 5] x 0 + [0 - 10][0 - 5] x 0.5
Cov(q, z) = (10)(5)(0.5) + (-10)(5)(0.5) + (10)(-5)(0) + (-10)(-5)(0.5)
Cov(q, z) = 25 + (-25) + 0 + 25 = 50
Therefore, the covariance of q and z is 50.
The correct option (B) is 100, so it seems there was an error in the given solution. The correct answer based on the information provided is 50.