If $X$ and $Y$ are two independent random variables, are $X^{2}$ and $Y$ independent too?

Question

Let $X \sim \mathrm{Unif }[0,1]$ and $Y\sim \mathrm{Gamma}(2,r)$, with $r>0$, be two independent random variables.

I have to find $\mathrm{Cov}(U,V)$ where $U = X$ and $V = XY$.

From previous points, I have calculated that $$f_{U,V}(u,v) = \frac{r^{2}ve^{-r\frac{v}{u}}}{u^{2}} $$ with $0 < u < 1 $ and $ v > 0 $,
and$$f_{V}(v) = re^{-rv}$$ with $v > 0$, hence, $V \sim \textit{Exp}(r)$.

Thus, \begin{align*} \mathrm{Cov}(U,V) &= \mathbb{E}(U\cdot V) - \mathbb{E}U \cdot\mathbb{E}V \\ &= \mathbb{E}(X\cdot XY) - \mathbb{E}X\cdot\mathbb{E}(XY) \\ &\overset{indep.}{=} \mathbb{E}(X^{2}\cdot Y) - (\mathbb{E}X)^{2}\cdot\mathbb{E}Y \end{align*}

• If $X^{2}$ and $Y$ are independent, then $$ = \mathbb{E}Y\cdot \textit{Var}(X)$$ which would be pretty convenient. But also how should I prove this?
• If they're not independent, then I'm wondering if there is an easy way to evaluate it, without having to find the density function of $UV$ which I predict would end up having the same "problem". I know that there is a way to approximate $\lvert\mathbb{E}(UV)\rvert$ (Cauchy-Schwarz inequality), that is, $$\lvert\mathbb{E}(UV)\rvert \leq \sqrt{\mathbb{E}(U^{2})\cdot\mathbb{E}(V^{2})}$$ but I'm not sure this is the way to evaluate this.

(I've searched around for an answer, but they all involve Measure Theory which I have no knowledge of).

Any suggestion/help would be appreciated.

Answer 1

The variables $X^2$ and $Y$ are independent (by definition) if for all $x,y$ we have $\Pr(X^2\leq x,Y\leq Y)=\Pr(X^2\leq x)\Pr(Y\leq Y)$. Clearly we can assume $x\geq 0$ here. So $$\Pr(X^2\leq x,Y\leq Y)=\Pr(-\sqrt x\leq X\leq \sqrt x,Y\leq y)\\ =\Pr(X\leq \sqrt x,Y\leq y)-\Pr(X\leq -\sqrt x,Y\leq y)\\ =\Pr(X\leq \sqrt x)\Pr(Y\leq y)-\Pr(X\leq -\sqrt x)\Pr(Y\leq y)\\ =[\Pr(X\leq \sqrt x)-\Pr(X\leq -\sqrt x)]\Pr(Y\leq y)\\ =\Pr(X^2\leq x)\Pr(Y\leq y).$$

Answer 2

For any continuous functions $f$ and $g$ (and, in fact, for any Borel measurable functions $f$ and $g$) the variables $f(X)$ and $g(Y)$ are independent. So it is certainly true that $X^{2}$ and $Y$ are independent.