Properties of variance

Question

I was doing the following question:

Let X be a random variable with Var(X) = 1. What can we say about the following(are they true or not)?

Var(X+1) = 2

Var(2X) = 4

Var(-X) = 1

$E[X^{2}]\ge 1$

I am not sure how I can make a judgement about any of this? Can someone prove their answers to me? As I have the answers but I am not sure how to either mathematically/intuitively makes sense?

Answer 1

Variance is not linear -- see https://stats.stackexchange.com/questions/184998/the-linearity-of-variance. For the first one, the question is whether $X$ and $1$ are independent (think about the definition). For the second one, we have $\text{Var}(2X) = 2^2 \text{Var}(X) = 4$. Similar ideas apply to the third one. Finally $$\text{Var}(X) = E(X^2) - E(X)^2 = 1 \implies E(X^2) = 1 + E(X)^2.$$ Notice $E(X)^2 \geq 0$, so $E(X^2) \geq 1.$

Answer 2

Hint: for random variables $X$ and $Y$ and real numbers $\alpha$ and $\beta$, you have the follow: $$ \text{Var}(\alpha X+\beta Y)=\alpha^2\text{Var}(X)+\beta^2\text{Var}(Y)+2\alpha\beta\text{Cov}(X,Y). $$ This should help you with the first 3. For the last one, try to use: $$ \text{Var}(X)=E(X^2)-[E(X)]^2. $$