I ran across this identity for a conditional PDF where the dependent random variable X is continuous and the independent variable N is discrete:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{P(x<X<x+dx|N=n)}{dx}=\frac{P(N=n|x<X<x+dx)}{P(N=n)}\frac{P(x<X<x+dx)}{dx}[/tex]

In the limit as dx approaches 0 this yields:

[tex]f_{X|Y}(x|n) =\frac{P(N=n|X=x)}{P(N=n)}f(x)[/tex]

I think I understand the 2nd step, but not the initial identity. The reversing of the conditions (from X dependent on N to N dependent on X) reminds me of Bayes Law, but if he is using Bayes Law here, it is not clear to me exactly how. Could someone help me understand this identity?

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# Mixed Conditional PDFs

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