Expected value and variance of a transformed random variable

Question

X is a binomial random variable with $n = 10000$ and $p = 60\%$ (hence $E(X) = 10000 * 60 = 6000$). Now Y is a transformed random variable with $Y = 100000 - 7X$.

I have to calculate the expected value $E(Y)$ and variance $V(Y)$ of Y.

For $E(Y)$ I thought of this: $E(Y) = E(100000 - 7X) = E(-7X) + 100000 = 100000 - E(7X) = 100000 - 7E(X) = 100000 - 7 * 6000 = 58000$

But how do I calculate $V(Y)$?

Answer 1

$$V(Y)=V(100000-7X)=7^2\cdot V(X)$$

I believe you can finish the exercise from here.

Answer 2

In general if $a,b$ are constants then: $$\mathsf{Var}(aX+b)=a^2\mathsf{Var}(X)$$ Verification: $$\mathbb E\left(aX+b-\mathbb E(aX+b)\right)^2=\mathbb E\left(aX+b-a\mathbb EX-b)\right)^2=\mathbb Ea^2(X-\mathbb EX)^2=a^2\mathbb E(X-\mathbb EX)^2$$