Let $X$ and $Y$ be random variables where $Y = cX$ for some constant $c > 0$. Prove that $$E(Y) = cE(x)$$ and $$V(Y) = c^2 V(x)$$ where $E(Y)$ is the expectation of $Y$, $E(x)$ is the expectation of $X$, and $V(Y)$ and $V(x)$ are the variance of $Y$ and $X$ respectively.
I have a bit of a concept as of why this works. What I was thinking is that since we know that the $E(Y)$ is equal to the sum of $f(k)P(x=k)$ where $f(k)$ is some function (here $Y = cX$)
From here though I do not know where to go to prove this. Any help is appreciated. Thank you.
You should first prove expectation is linear (proof is in the discrete case, but in the general case, its linearity of integrals, since expectation is just an integral).
Then, by linearity, prove $var(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2$ (there is some algebra in the middle). Then, apply linearity and the definition of $Y$ to this to get the result.