expectation of $X,Y$ RV's and $Y>0$

Question

suppose I have 2 independent random variables $X,Y$, and $Y>0$, also suppose that $Z=\frac{X}{Y}$ and $\mathbb{E}[Z]$ exists and $0<\mathbb{E}[Z]<1$ what of the following has to be correct?

1. $\mathbb{P}[X\leq Y]=1$
   
2. $\mathbb{P}[0\leq X]=1$
   
3. $\mathbb{P}[0\leq X\leq Y] > 0$
   
4. $0\leq \mathbb{E}[X] \leq \mathbb{E}[Y]$
   

how can i even approach this question? i've tries using chebichev and markov inequalities but nothing came out of it.

Answer 1

4. is correct. Actually denoting $a= E(Y)$ and $b=E(\frac{1}{Y})$ we observe that $$0\leq E\left(\left(\sqrt{\frac{Y}{a}}-\sqrt{\frac{Y^{-1}}{b}}\right)^2\right)$$ implies that $1\leq ab$ (this is Schwarz inequality) therefore $E(|Z|)<1$ implies $$E(|X|) \leq \frac{1}{E(\frac{1}{Y})}\leq E(Y).$$