Let (, Pr) be a probability space. Consider two independent random variables X and Y such that Var[X] = 9 and Var[Y] = 16. Calculate the standard the deviation of 2X + 3Y. For this problem, you may state without proof that if X and Y are independent, then 2X and 3Y are also independent.

solution Since, X and Y are independent, we can use the fact that the variance of a sum of independent random variables is the sum of their variances: Var[2X + 3Y] = Var[2X] + Var[3Y] Simplifying further we get: Var[2X + 3Y] = 4 Var[X] + 9 Var[Y] Substituting the given variances for X and Y: Var[2X + 3Y]= 4(9) + 9(16) = 180 The standard deviation is the square root of the variance, which is 180 = 65

Is it the right way to solve it?