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Are these two joint probabilities always equal: P(a,b) = P(b,a) ? Or does it matter the order a,b vs b,a?

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  • $\begingroup$ Suppose a is a set of weights and b a set of heights of subjects, so that an expression like P(a,b) usually refers to the chance that a weight is in a at the same time a subject's height is in b. Could you please explain to us what P(b,a) could possibly mean? Literally, it refers to the chance that a subject's weight is in the set b of heights and her height is in the set a of weights! Or, are you perhaps using the comma as a shorthand for the intersection of two events concerning the same quantity (so that, for instance, both a and b are sets of weights)? $\endgroup$
    – whuber
    Sep 22 '20 at 16:51
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If $a$ and $b$ are events, yes, they're equal. But, in general events are denoted with uppercase letters, so I'm assuming they are specific real values. If that's the case, it is an abuse of notation. Normally, it should be $P(A=a,B=b)$, which is equal to $P(B=b,A=a)$. Here, $\{A=a\}$ and $\{B=b\}$ are events, and we can change their ordering because the comma sign basically mean $\cap$.

If you're asking for PDFs and PMFs, they're commonly denoted with lowercase $p$ or $f$, and commonly with a subscript to denote the ordering of RVs. For example, $p_{A,B}(a,b)$, and surely $p_{A,B}(a,b)\neq p_{A,B}(b,a)$.

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  • $\begingroup$ Thanks, in my case a and b are both events and random variables. Can you please elaborate on why pA,B(a,b) different from pA,B(b,a)? $\endgroup$ Sep 22 '20 at 14:03
  • $\begingroup$ Random variable being equal to a specific value is an event, but the RV itself is not an event. So, you should clarify what $a,b$ is. As per the inequality, $a,b$ denotes specific values there, like $1,2$ and $A,B$ are RVs. $\endgroup$
    – gunes
    Sep 22 '20 at 14:04
  • $\begingroup$ I see, a,b are events in my case. But now I am confused, what's the difference between p(A=a,B=b) and pA,B(a,b)? $\endgroup$ Sep 22 '20 at 14:09
  • $\begingroup$ If $A,B$ are discrete RVs, there is no difference. $\endgroup$
    – gunes
    Sep 22 '20 at 14:11
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    $\begingroup$ Thanks very much! $\endgroup$ Sep 22 '20 at 15:27
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They are the same. Taking the simpler example of discrete random variables $A$ and $B$, $P(a,b)$ is just a short notation for $P(A=a \wedge B=b)$, which is obviously the same as $P(B=b \wedge A=a)$.

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  • $\begingroup$ (+1) It's correct based on the assumption that $a,b$ are specific values corresponding to RVs $A,B$, but the OP had little details clarifying this. $\endgroup$
    – gunes
    Sep 22 '20 at 14:20
  • $\begingroup$ Thanks, this is correct as well. $\endgroup$ Sep 22 '20 at 15:29

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