Let X be a random variable uniformly distributed over an interval [0,6]. Let Y be a random variable uniformly distributed over an interval [2,14]. It is known that X and Y are independent.
(a) Find mean and variance of 3X−Y−4.
The mean should be: E(3X−Y−4)=3E(X)−E(Y)−4.
I have found that; E[X] = 3, and E[X^2]=13, E[Y] = 8, and E[X^2]=78. Thereby, 3E(X)−E(Y)−4 = 3*3-8-4 = -3, so the mean of the expression is -3.
I am not sure how to find the variance of the whole expression, but I have found the variance of X = E(X^2)-(E(X))^2 = 13-9 = 4, and Y = = E(Y^2)-(E(Y))^2 = 78-64 = 14.
Var(3X)-Var(Y) = 9Var(X)+Var(Y) = 9*4+14 = 50, so the variance of the expression is 50 and the mean of the expression is -3.