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I came across this information in Statistical Inference by Casella and Berger. The problem was the authors didn’t explain how they derived the second equality (highlighted). I never saw such an expression as $P(X > s, X > t)$. Also, why was it divided by $P(X > t)$? Was it because this was the case of conditional probability? Could you please tell me the logic of that derivation? Thanks.

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You are probably familiar with the rule and just don't recognize it in this setting. It applies to densities...

$f(x|y)=\frac{f(x,y)}{f(y)}$

...but also to sets (which is your case)...

$P(A|B)=\frac{P(A \cap B)}{P(B)}$

In words, a conditional thing equals the joint thing over the marginal of the condition. This follows from the "chain rule".

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  • $\begingroup$ Thanks, Alvaro. To clarify, did the rule as mentioned in your first sentence refer to conditional probability? If so, I wonder why the authors didn't say it, and should had used the expression $P(A \cap B)$ instead? $\endgroup$
    – Nemo
    Commented Jan 23, 2020 at 12:03
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    $\begingroup$ Yes, it's a factorization of conditional probability. $P(A\cap B)$ is set notation and $P(A,B)$ is probability notation (I fixed that in my answer now, sorry), but they signify the same thing. $\endgroup$
    – suckrates
    Commented Jan 23, 2020 at 12:10

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