Which of the following are always true for random variables $X$, $Y$ and real numbers $a$, $b$?

Question

Which of the following are always true for random variables $X$, $Y$ and real numbers $a$, $b$?

1. The variance of $X$ is always non-negative.
2. The standard deviation of $X$ is always non-negative.
3. If $V(X)=V(Y)$, then $V(X+a)=V(Y+b)$.
4. If $V(aX)=V(bX)$ for $a≠0$ and $b≠0$, then $a=b$.
5. If $E[X]=E[Y]$ and $V(X)=V(Y)$, then $X=Y$.
6. If $E[X]=E[Y]$ and $V(X)=V(Y)$, then $E[X^2]=E[Y^2]$.

I know that 1 and 2 are always true since variance can only be 0 (if all values are the same) or positive (due to the squaring).

I am certain that option 3 is false as $a$ and $b$ may differ. Option 4 should be also true since they multiply the same value, so if the result is the same $a$ and $b$ must equal.

The last 2 options baffle me as it includes Y as well.

Answer 1

3) $V(X+a)=V(X)=V(Y)=V(Y+b)$ so the statement is true.

4) if $V(X)=0$ then $V(aX)=a^2V(X)=0=b^2V(X)=V(bX)$ for all constants $a,b$, so the statement is false. It is false also under the extra condition that $V(X)>0$. Then equality $V(X)=V(-X)$ provides a counterexample.

5) False. A random variable is not determined by expectation and variance. Even stronger: also the distribution of a random variable is not determined by expectation and variance.

6) This statement is true because $\mathbb EX^2=V(X)+(\mathbb EX)^2=V(Y)+(\mathbb EY)^2=\mathbb EY^2$.

Answer 2

Number $5$ is false, it is not sufficient to have common expectation and variance to conclude that the random variables are the same.

Number $6$ is true due to the formula $Var(X)=E(X^2)-E(X)^2$. It can be found here https://en.wikipedia.org/wiki/Variance