Question

Suppose that X is a random variable with mean 30 and standard deviation 10. Also suppose that Y is a random variable with mean 20 and standard deviation 8. Assume that the correlation between X and Y is 0.4. Find the mean and standard deviation of the random variable Z for each of the following cases. Be sure to show your work. a. Z= 2 + 10X. b. Z = X + Y. c. Z = −5X − 3Y

Answer 1

Part a: For variable X using z score you have (x-30)/10, replace X for this expression in (a), i mean Z=2+10((x-30)/10) = 2+x-30= put this in the way of z score = (x-18)/1, so the mean is 18 and standard deviation is 1.
Part b: Z=X+Y= (x-30)/10 + (x-20)/8 = solve adding fractions = (18x-440)/80 now divide all the numbers between 18 (because you need to show as z score way) = (x-24.4)/4.4, so the mean is 24.4 and standard deviation is 4.4.
For part c,do the same.