How to calculate $\text{Var}(2X)$ where $X$ is a random variable?

Question

Roll a fair die and let $X$ be the number of dots showing on top. What are $\text{E}(2X)$ and $\text{Var}(2X)$?

Answer 1

Notice that:

$$\text{Var}[X] = \mathbb{E}[(X-\mathbb{E}[X])^2].$$

Consider $Y = 2X.$ Then:

$$\mathbb{E}[Y] = \mathbb{E}[2X] = 2 \mathbb{E}[X].$$

Moreover:

$$\text{Var}[Y] = \mathbb{E}[(Y-\mathbb{E}[Y])^2] = \mathbb{E}[(2X-2\mathbb{E}[X])^2] = \mathbb{E}[4(X-\mathbb{E}[X])^2] = 4 \text{Var}[X].$$

In general:

$$\text{Var}[aX] = a^2 \text{Var}[X].$$

---

Addedum

Recall that $\mathbb{E}[aX] = a \mathbb{E}[X]$.