see questions: correlations, covariance, sum of normal

please make sure to well explain your answers!

Thanks

Let X and Y be independent normal random variables. a) Is 2X 4 always normal? False a b) Is 3X 4Y always normal? True 8 c) Is X2 + Y always normal? False a \fAs in the preceding example, let Z, V, and W be independent random variables with mean 0 and variance 1, and let X = Z + V and Y = Z + W. We have found that p (X, Y) = 1/2. a) It follows that: P ( X , - Y ) = P ( - X, - Y) = b) Suppose that X and Y are measured in dollars. Let X' and Y' be the same random variables, but measured in cents, so that X' = 100X and Y' = 100Y. Then, P ( X' , Y ') = c) Suppose now that X = 3Z + 3V + 3 and Y = -2Z - 2W. Then P ( X , Y ) = d) Suppose now that the variance of Z is replaced by a very large number. Then p (X, Y) is close to e) Alternatively, suppose that the variance of Z is close to zero. Then p (X, Y) is close to