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I have found this equality online and I am struggling to understand how/why it holds.

$$ \int_{-\infty}^\infty \int_{-\infty}^{y+1} f_{(X\mid Y)}(x\mid y)\,dx\,dy = \int_{-\infty}^\infty \mathbb P(X<y+1 \mid Y=y)\,dy$$

Is this correct? If so, how come?

StubbornAtom
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user745587
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1 Answers1

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If $f_{(X|Y)}(\cdot| y)$ is the density of $P(\cdot | Y=y),$ then $$\int_{- \infty}^{y+1} f_{(X|Y)}(x|y) dx = P(X \leq y+1 | Y=y)$$ follows by definition. See e.g. here for a very good description.

blat
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