Question

A random variable X is normally distributed with a mean of 121 and a variance of 121, and a random variable Y is normally distributed with a mean of 150 and a variance of 225. the random variables have a correlation coefficient equal to 0.6. Find the mean and variance of the random variable :
W= 6X + 3Y

Answer 1

Consider the given problem here “X” and “Y” be two random variables follows normal distribution and their mean and variance is also given in the question.

Now, the correlation coefficient between “X” and “Y” is “0.6”, “V(X)=121” and “V(Y)=225”. So, the standard deviation of “X” and “Y” are “?x = 11” and “?y = 15” respectively.

=> Cov(X, Y) = ?*?x*?y = 0.6*11*15 = 99.

=> the mean of “W” is “E(W) = 6*E(X) + 3*E(Y) = 6*121 + 3*150 = 1176, => E(W) = 1,176.

Now, the variance of “W” is given below.

=> V(W) = 6^2*V(X) + 3^2*V(Y) + 2*6*3*Cov(X, Y) = 36*121 + 9*225 + 36*99 = 9,945.

=> V(W) = 9,945. So, “W” is a random variable follows normal distribution with mean “1176” and variance “9945”.