Is the joint probability $p(X=x,Y=y)$ equivalent to $p(X=x \cap Y=y )$? If it is, why do we use two different notations?


Yes, they mean precisely the same thing.

Why different notations? Well, this is not the only place in mathematics where there are multiple notations. For example, $A'$, $A^c$, and $\bar{A}$ are all used for the complement of $A$.

The version with the commas is more compact, particularly since the other version should really read $\Pr((X=x)\cap (Y=y))$. Think of the trees saved.

The version $\Pr((X=x)\cap (Y=y))$ emphasizes the logical structure, so has some pedagogical advantages.

  • $\begingroup$ Thanks. What is the meaning of $(X=x)\cap(Y=y)$ when X is (say) a number of coin flips and Y is the color of the sky? I don't understand the meaning of an intersection of two events defined on different sample spaces. $\endgroup$ – usual me Oct 1 '13 at 2:56
  • 2
    $\begingroup$ The sample spaces are not different. Let us say that the colour of the sky can be B(lue) or G(ray). Then the sample space is the collection of ordered pairs $(n,C)$ where $n$ is a number and $C$ is one of B or G. $\endgroup$ – André Nicolas Oct 1 '13 at 3:21
  • $\begingroup$ Let S be the collection of ordered pairs $(n,C)$. My understanding is that $(X=x)$ is an event over S, and is equivalent to $\{(x,B),(x,G)\}$. Is this correct? $\endgroup$ – usual me Oct 1 '13 at 6:52
  • $\begingroup$ Yes, that's right. Formally, we have to go to the "product space," and that is how the informal event $X=x$ is represented. $\endgroup$ – André Nicolas Oct 1 '13 at 7:02

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