Definition of joint probability distribution function (PDF), why the integrals are flipped?

Question

From this note, on page 17, it says:

Observe that the inner integral is from $-\infty$ to $y$, but with respect to $dx$. Should it make sense? The integral seems to be flipped. If I am integrating along $x$, my limit of integration should be $x$ instead of $y$?

This Wikipedia article seems to agree with me: https://en.wikipedia.org/wiki/Multiple_integral#Integrating_constant_functions

The immediate paragraph confirms my suspicion:

Observe now the $dx$ is with taking the limit from $-\infty$ to $x$. This seems correct to me.

Can anyone confirm if there is a typo in the notes or if the limit of integration of the first integral makes sense. If it makes sense, how does it explain the limit of integration of the second integral?

Answer 1

It's a typo.

We have $$F_{X,Y}(\color{blue}{x},\color{blue}y) = \int_{-\infty}^\color{blue}y \int_{-\infty}^\color{blue}{x} f_{X,Y}(\color{purple}u,\color{purple}v) \, d\color{purple}u \,d\color{purple}v.$$

Answer 2

The integral could have been written more clearly as $$F_{X,Y}^{\,}(x,y)=\int_{y'=-\infty}^{y'=y}\int_{x'=-\infty}^{x'=x} f_{X,Y}^{\,}(x',y') \, dx' \, dy'$$ or as $$F_{X,Y}^{\,}(x,y)=\int_{x'=-\infty}^{x'=x}\int_{y'=-\infty}^{y'=y} f_{X,Y}^{\,}(x',y') \, dy' \, dx'$$