Let the total score for the quiz combine L and H in a 1 to 2 weighting (the better score is worth more). So, T = L + 2H. This total score T is out of 54 points. Determine the expected value and variance of T.
So in a previous problem I've found the Expected Value of of T when it is out of 36 points, and I know you can scale 3/2 * E(T) of 36 points to find the Expected Value out of 54 points.
How would scaling affect the Variance of T? When calculating variance would I need to go through the entire PMF with the scaled values in place or can I also multiply V(X+Y) = V(X) + 2*(V(Y)) + 2*COV(X,2Y)?
Note that if $X$ is a random variable with a finite second moment and expected value $m$, and $k \in \mathbb{R}$, then $\mathbb{E}[kX] = km$ by linearity of expectation. Further, $$ \mathbb{Var}(kX) = \mathbb{E} \left[ \left( kX - \mathbb{E}[kX] \right)^2 \right] = \mathbb{E} \left[ \left( kX - km \right)^2 \right] = k^2 \mathbb{E} \left[ (X - m)^2 \right] = k^2 \mathbb{Var}(X). $$