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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?

usual me
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1 Answers1

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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.

André Nicolas
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  • 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. – usual me Oct 01 '13 at 02:56
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    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. – André Nicolas Oct 01 '13 at 03:21
  • 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? – usual me Oct 01 '13 at 06:52
  • Yes, that's right. Formally, we have to go to the "product space," and that is how the informal event $X=x$ is represented. – André Nicolas Oct 01 '13 at 07:02
  • If x={1,2}, Y={B,G} then the product space is { (1,B)......} Then P(X=1, Y=B) = 0.25. Where is the intersection..I can’t imagine it as an intersection. Could you please help me – Shashaank Oct 26 '20 at 16:05