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As shown in the image below, how does A turn into B? What is the intuitive logic behind the disappearance of sigma y? Thanks!!

enter image description here

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You've "summed out" all dependence on $y$, essentially. Try looking at a specific example. – icurays1 Dec 2 '12 at 3:19
This is exactly correct. Formally this is called the Law of Total Probability. – Patrick Dec 2 '12 at 3:48

The event $\{X=x\}$ is the union of the mutually exclusive events $\{X=x,\ Y=y\}$ for all $y$.

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