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I've two independent and uniform random variables $Z_1,Z_2 \in \lbrace 1,2,3,4 \rbrace$. Can you tell me why the following equality is correct please?

$$ P(Z_1+Z_2=2,Z_1=1,Z_2=1)=P(Z_1=1,Z_2=1) $$

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  • $\begingroup$ The LHS is vague, what does it mean? $\endgroup$
    – gunes
    Jan 8, 2022 at 8:58
  • $\begingroup$ Hello @gunes do you mean that the notation is vague? $\endgroup$ Jan 8, 2022 at 8:59
  • $\begingroup$ Yes @GennaroArguzzi $\endgroup$
    – gunes
    Jan 8, 2022 at 8:59
  • $\begingroup$ @gunes sorry there was a typo $\endgroup$ Jan 8, 2022 at 9:00

1 Answer 1

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This holds even if the variables are neither independent nor uniform because the event $Z_1+Z_2=2$ does not add anything when the RVs are both $1$.

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  • $\begingroup$ Yes, similar to that. $\endgroup$
    – gunes
    Jan 8, 2022 at 9:04
  • $\begingroup$ is it like writing the same event twice? $\endgroup$ Jan 8, 2022 at 9:04
  • $\begingroup$ thank you very much $\endgroup$ Jan 8, 2022 at 9:04

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