A person is traveling between two places, and has 3 options for transportation. The jth option would take an average of µj hours, with a standard deviation of $\sigma_j$ hours. The person randomly chooses between the 3 options, with equal probabilities. Let T be how long it takes for him to get from place 1 to place 2.
(a) Find E(T). Is it simply (µ1 +µ2 +µ3)/3, the average of the expectations?
Expectation is additive, there is equal probabilities between options, so I agree it is the "average" of the averages.
(b) Find Var(T). Is it simply ($\sigma_1^2+\sigma_2^2+\sigma_j^2$)/3, the average of the variances?
Variance is additive when you are adding iid E(T)'s (confirmed: https://apcentral.collegeboard.org/courses/ap-statistics/classroom-resources/why-variances-add-and-why-it-matters), but I am unsure if you just "average" the variances. Guidance?