Adding Independent Random Variables Given Their Individual Expectations and Variance

Question

How do I add or subtract independent random variables (R.Vs) when given their individual expectations and variance? I'm a student in high school and I haven't covered distributions yet, so please try not to use them.

Example, R.Vs A, B & C

Where

$E(A)= 35\;\;\ Var(A)=8\\ E(B) = 25\;\;\;\; Var(B)=9\\$

Calculate the expectation and variance of:

$A + 2B$

Answer 1

Let $A$ and $B$ be two random variables and $c$ be a constant. Then,

1. $\mathbb{E}[A + cB] = \mathbb{E}[A] + c\mathbb{E}[B]$ and
2. $\operatorname{Var}(A + cB) = \operatorname{Var}(A) + c^2 \operatorname{Var}(B)$ (assuming $A$ and $B$ are independent).

Variance is defined in terms of the expectation. In particular, $\operatorname{Var}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2]$. See if you can use this definition to prove property (2) from property (1).