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What are the assumptions that I need on the function $P( X | Y = y )$ (where $P$ is the conditional probability density function of X knowing an observation y of Y - so is bounded by one and always positive) in order to be able to write the following?

$$ \frac{\partial }{\partial \tau} \left( \int_{-\infty}^{\tau} P( X = x | Y = y ) \right) = P( X = \tau | Y = y ).$$

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The partial derivative $$\frac{\partial}{\partial x}\int_{-\infty}^{x}f_{X|Y}(x^\prime|y) \mathrm{d}x^\prime=P(X=x|Y=y)=f_{X|Y}(X=x|Y=y)$$ always if the conditional distribution $F_{X|Y}(x|y)=\int_{-\infty}^{x}f_{X|Y}(x^\prime|y) \mathrm{d}x^\prime$ is differentiable at every $X=x, \forall y \in \Omega_y$; which simply means that the Conditional CDF has to be well defined to be differentiable everywhere. At those places where the derivatives do not exist, the distribution $f_{X|Y}$ cannot be defined as we want to and hence does not exist.

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  • $\begingroup$ I understand, so there is no need to consider the fact that the integral is from $-\infty$ and so there is a limit in there? $\endgroup$
    – Matteo
    Oct 17, 2013 at 7:38
  • $\begingroup$ No it's totally ok because any PDF to be valid should have $\int_{-\infty}^{+\infty}f_X \mathrm{d}x=1$. i.e. The real line is the support for any PDF where in $f_X$ can be zero at some places or can even be left undefined (which will lead to weird cases like the Random variable cannot take that value at all). $\endgroup$
    – Sudarsan
    Oct 17, 2013 at 19:43
  • $\begingroup$ It boils down to the Differentiability of the CDF $F_X(x)$. If it is continuous and differentiable everywhere $f_X(x)$ is well defined. If $F_X(x)$ is continuous but not differentiable at a few places, then at those places the PDF is undefined i.e. $X$ cannot take those values at all. If $F_X(x)$ is not continuous at a place then it means that there is a probability mass for $f_X(x)$ at that place and it is defined using the well known $\delta$ function; so it's still a well defined point in terms of the PDF. $\endgroup$
    – Sudarsan
    Oct 17, 2013 at 19:47

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