Question 1207202
Understanding the Problem:

We are given two independent random variables, X and Y, with their respective means and variances:

E[X] = 4
E[Y] = 5
Var[X] = 1
Var[Y] = 2
Key Property of Independent Random Variables:

For independent random variables X and Y, the following properties hold:

Expected Value of the Sum: E[X + Y] = E[X] + E[Y]
Variance of the Sum: Var[X + Y] = Var[X] + Var[Y]
Expected Value of the Product: E[XY] = E[X] * E[Y]
Variance of the Product: Var(XY) = E[X^2]E[Y^2] - (E[X]E[Y])^2
Solving the Problem:

A. E[4X + 2Y] = 4E[X] + 2E[Y] = 44 + 25 = 26

B. E[5X - Y] = 5E[X] - E[Y] = 5*4 - 5 = 15

C. Var[3X + 2Y] = 9Var[X] + 4Var[Y] = 91 + 42 = 17

D. Var[5Y - 3X] = 25Var[Y] + 9Var[X] = 252 + 91 = 61

Therefore, the answers are:

A. 26
B. 15
C. 17
D. 61