Dependency and correlation between a random variable and it's square

Question

I have the following question. At first this seemed very silly but after thinking about it, I found my self struggling.

Given $X$, a random variable, I should decide if the following sentences are right or wrong and explain why:

1 - $X$ and $X^2$ are always independent.

2 - $X$ and $X^2$ are never independent.

3 - $X$ and $X^2$ are always correlated.

4 - $X$ and $X^2$ are never correlated.

Here are my answers. I was able (I think) to answer dependency part, but the correlation part I couldn't.

1 - False: since $X^2$ could only be achieved with $X$ and $-X$, it is dependat with $X$.

2 - False: As said before, we can get $X^2$ using $-X$. This way $X^2$ and $X$ are independant.

3 & 4 I don't have a clue.

Could you guys help me with it? Did I answer questions 1 and 2 right? any explenation by words or via mathemtical approach would help.

Thank you!

Answer 1

Just to spell it out in more detail than the comments, here's one counterexample for 3. Let $ X \sim U[-1,1] $ and let $ Y = X^2 $. Then $$ \begin{align*} \mathbb{C}\text{ov}(X,Y) &= \mathbb{E}[X^3] \\ &= \int_{-1}^1 \frac{x^3}{2} dx \\ &= 0. \end{align*} $$ For question 1, you can basically use any example where squaring would lead to a constant. Remember that a constant is independent of all random variables, including itself.

Answer 2

Since this is a self-study problem, I will only give some helpful hints and leave it to the OP to fill in the gaps (how to do the math).

1. $X$ and $X^2$ need not be independent. Consider the image below comparing $X\sim U(0,1)$ with $X^2$. The pattern shows clear structure.

set.seed(2023)
N <- 1000
x <- runif(N, 0, 1)
plot(x, x^2)

2. However, $X$ and $X^2$ can be independent, such as if $X$ is a degenerate distribution with all probability mass on one value.

3. $X$ and $X^2$ do not have to be correlated, as the image below shows for a random variable $X$ with uniform probability mass on $-1$, $0$, and $1$.

set.seed(2023)
N <- 1000
x <- c(-1, 0, 1)
plot(x, x^2)

4. However, $X$ and $X^2$ can be correlated, as the first example shows.

You have the answers. Now it is up to you to prove my claims using the math instead of just getting intuition from pictures. For the independence claims, rely on the definition of independence, $P(A\cap B) = P(A)P(B)$ for all events $A$ and $B$, and either find $A$ and $B$ such that the equation fails (to show dependence) or show that the equation holds for all events $A$ and $B$. For the correlation claims, rely on the definition of covariance as $\mathbb E\left[ (X-\mathbb E\left[X\right])(Y-\mathbb E\left[Y\right) \right]$, knowing that for distributions with positive variance like I have given in 3 and 4, nonzero covariance corresponds to nonzero correlation.