1
$\begingroup$

Consider two real-valued random variables $Y_1: \Omega \rightarrow \mathbb{R}$ and $Y_2:\Omega \rightarrow \mathbb{R}$. Even if they have the same domain and codomain, $Y_1 $ and $Y_2$ generate different $\sigma$-algebras, $\sigma(Y_1)$ and $\sigma(Y_2)$, and are defined in different probability spaces, $(\Omega, \mathcal{F}_{Y_1}, \mathbb{P}_{Y_1})$ and $(\Omega, \mathcal{F}_{Y_2}, \mathbb{P}_{Y_2})$. Suppose that $Y_1$ and $Y_2$ are identically distributed: $\mathbb{P}_{Y_1}(Y_1 \in E)=\mathbb{P}_{Y_2}(Y_2 \in E)$ $\forall E \in \mathcal{B}(\mathbb{R})$, i.e. $\mathbb{P}_{Y_1}(\omega \in \Omega | Y_1(\omega) \in E)=\mathbb{P}_{Y_2}(\omega \in \Omega | Y_2(\omega) \in E)$ $\forall E \in \mathcal{B}(\mathbb{R})$.

Could you help me in saying whether the following statements are true?

(1) Identical distribution does not imply that $\mathbb{P}_{Y_1}$ and $\mathbb{P}_{Y_2}$ are the same functions because the sets $\{\omega \in \Omega | Y_1(\omega) \in E \}$ and $\{\omega \in \Omega | Y_2(\omega) \in E \}$ could be different.

(2) Identical distribution does not imply that $Y_1$ and $Y_2$ are the same function, i.e. $Y_1(\omega)=Y_2(\omega)$ $\forall \omega \in \Omega$. Therefore, even if $Y_1$ and $Y_2$ are identically distributed, $\sigma(Y_1)$ and $\sigma(Y_2)$ are different.

(3) Identical distribution implies that the image of $Y_1$ and $Y_2$ is the same.

$\endgroup$
  • 1
    $\begingroup$ This seems like routine bookwork, perhaps assigned work for some subject, or perhaps just for your own personal study. Either way, could you please add the self-study tag, and read the guidelines at the link I gave above, then edit your question to accord with those guidelines. $\endgroup$ – Glen_b -Reinstate Monica Mar 20 '14 at 9:17
1
$\begingroup$

Think about which information of a random variable will be hidden by the distribution function. So the answers are yes, yes, and almost everywhere the same.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.