Show that the events $\{X ≤ a\}$ and $\{Y ≤ b\}$ are independent.

Question

Suppose that $X$ and $Y$ are real-valued random variables and suppose that for all $a$ and $b$ in $R$, the events $\{X ≤ a\}$ and $\{Y ≤ b\}$ are independent. Prove that $X$ and $Y$ are independent.

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I know by definition if $\sigma(X)$ and $\sigma(Y)$ are also independent then $X$ and $Y$ are independent. How this will help me to confirm that the events $\{X ≤ a\}$ and $\{Y ≤ b\}$ are independent?

Answer 1

Without knowing what definition is known about independence of $X$ and $Y$, it is difficult to answer the question.

I know by definition if $\sigma(X)$ and $\sigma(Y)$ are also independent then $X$ and $Y$ are independent.

• So what can you tell about $\sigma(X)$? Do know the definition of $\sigma(X)$?
• Do you know what does it mean by independence of $\sigma(X)$ and $\sigma(Y)$?
• Do you know what family of sets can generate $\sigma(X)$.

If you are learning probability theory, these should all have answers/hints in your textbook.