Regarding the distribution of (Y1 + Y2 + Y3) - (X1 + X2 + X3 + X4). The answer sheet states N ~ (3*1200 - 4*800, 3*300^2 + 4*100^2). I expected it to be N ~ (3*1200 - 4*800, 3^2*300^2 + 4^2*100^2) due to the dimension of the variance. Please advice. Thanks!
Suppose \(X_1\) and \(X_2\) are independent and identically distributed as \(X\) The short answer is that \(X_1 + X_2 \neq 2X\). Think about it - rolling two dice and summing the score (outcomes: 2, 3, 4, ...., 12) is NOT the same as rolling one die and doubling it (outcomes 2, 4, 6, 8, 10, 12). So the variances will not be the same. \(var(X_1)+var(X_2) = 2var(X)\) whereas \(var(2X) = 2^2var(X)\)