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Does the following expression have a meaningful/intuitive interpretation in statistics?

$$\int_{-\infty}^x f_{X,Y}(u,y)du$$ where $f_{X,Y}$ is the joint density of $X$ and $Y$.

If we solve $\int_{-\infty}^x \int_{-\infty}^{\infty} f_{X,Y}(u,y)dydu$ (1) we can think about it in the following way:

  • First, we solve for the marginal density of $X$ (the inner term): $f_X= \int_{-\infty}^{\infty} f_{X,Y}(u,y)dy$.
  • Then, we integrate from $-\infty$ to $x$ to get $P(X \leq x) = \int_{-\infty}^x f_{X}(u)du$.

Is there a similarly nice way of breaking down the steps if we solve the problem in the following format: $\int_{-\infty}^{\infty} \left[ \int_{-\infty}^x f_{X,Y}(u,y)du \right] dy$ (i.e. apply Fubini's theorem to (1))?

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    This might belong better on the cross-validated stack exchange, focussing on statistical topics. – 2017-01-03
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    So it can be interpreted as a partial derivative of a joint CDF. So it is closely related to conditional CDF, or similar stuffs: http://math.stackexchange.com/questions/646449/conditional-cdf-from-joint-cdf-using-partial-derivatives – 2017-01-04
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    This is $f_Y(y)\cdot P(X\leqslant x\mid Y=y)$. – 2017-01-04

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