# Calculating Standard Deviation on 2 Independent Random Variables and More

This is the problem I was assigned in my homework:

I read in the textbook and believe that this method should be applied to solve the problem when dealing with 2 independent variables.

As a result, I've done so hand-written work and filled in the formula like this:

var(aX + bY) = 48^2 (1) + 6^2 (.0625)


However, as I have found incredibly weird and most likely wrong, the variance came out to be 2306.25, and thus the calculated standard deviation was 48.02. This seems wrong to me, but I can't lay my finger on it.

Explanation of variables and why I chose this formula

I'm on part A, and I calculated the expected amount of ice cream to be served in total to be 48 + 3 scoops of a mean of 2 oz each = 54 oz.

I went on to the second part of A, and came up with the formula above. Is there something wrong with how I set it up?

Edit: On Part B now

Expected value of ice cream being left in the box is: 48 - 2 = 46 oz. Variance formula: var(aX + bY) = 1^2 * 1 + 1^2 * .0625 = 1.0625 = 1.03

For the first part of (a), you probably should have uses $a=1$ ("one box") and b=3 ("plus three scoops from a second box") in the combined variance formula. But if the scoops are independent of each other and the box then you should be using a formula more like $Var(X+Y_1+Y_2+Y_3) = Var(X)+Var(Y_1)+Var(Y_2)+Var(Y_3)$ so here $1+0.0625+0.0625+0.0625$ [edited - see OpalE's comment]