Given two jointly distributed random variables $X, Y$, I would like to compute the conditional cumulative (or cumulative conditional?) probability $P(X \leq x | y)$ that given a value $y$ of $Y$, $X$ is less or equal than $x$. Is the following correct? \begin{align} P(X \leq x | y) &= \int_{X \leq x} \text{d}x' \, \frac{P(x', y)}{P(y)} \\ &= \frac{1}{P(y)} \int_{X \leq x} \text{d}x' \, \frac{\partial^2}{\partial x' \partial y} P(X \leq x', Y \leq y) \\ &= \frac{1}{P(y)} \frac{\partial}{\partial y} \int_{X \leq x} \text{d}x' \, \frac{\partial}{\partial x'} P(X \leq x', Y \leq y) \\ &= \frac{1}{P(y)} \frac{\partial}{\partial y} P(X \leq x, Y \leq y). \end{align} I.e., can I express $P(X \leq x | y)$ as a derivative of $P(X \leq x, Y \leq y)$ with respect to $y$ as I did above?


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