I am having trouble using what is given to prove the following. I think I have figured out a way to prove this using the variance shortcut formula, but that isn't what's being asked for I think. Does anyone have any ideas? Thanks!
Use the definition in Expression 3.13 to prove that $V(aX+b)=a^2\sigma_x^2$. [Hint: With $h(x)=aX+b$, $E[h(X)]=a\mu+b$ where $\mu=E(X)$. Expression 3.1 is:
$$ V[h(x)]=\sigma_{h(x)}^2=\Sigma[h(x)-E[h(x)]]^2\cdot p(x) $$
I don't quite understand, doesn't your expression 3.13 give you the answer you are searching for in the first place?
Regardless, you could use the fact that Var is both invariant under a change in location parameter, i.e. $Var(X + b) = Var(X)$ and the fact that $Var(aX) = a^2 Var(X)$. These follow by nice properties of the expectation, and noticing that $Var(X) = E[X^2] - E[X]^2$. It should be possible to see these by expanding out the definition of $E[\cdot]$.