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I'm confused about how a formula was found in Elements of Statistical Learning, pg. 11.

$$ EPE(f) = E(Y - f(X))^2 = \int [y - f(x)]^2 Pr(dx, dy) $$

$Pr(dx, dy)$ is the joint probability of $dx$ and $dy$ but how can we find probabilities of infinitesimals? Also, how did they find that integral?

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  • $\begingroup$ Please add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. $\endgroup$
    – gung - Reinstate Monica
    Commented Jan 14, 2016 at 19:18
  • $\begingroup$ You will also need to provide more context so that this question will stand on its own. In the future, people may not know what "ESL" is or have access to it. $\endgroup$
    – gung - Reinstate Monica
    Commented Jan 14, 2016 at 19:19

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The notation $Pr(dx, dy)$ is likely meant to cover both discrete and continuous distributions. In the continuous case, read this as $$Pr(dx, dy)=f(x,y)dxdy$$ The integral follows from the definition of an expected value of a function of random variables, see e.g. here. For example, in the univariate continuous case: $$ E[g(X)]:=\int g(x)f(x)dx $$

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