Help on Basics of Univariate and Bivariate Density and Distribution Functions First of all let me apologise if my MathJax could be improved.My ability in Mathematics extends to A-Level only, sometimes I just like to remind myself of the basics.
I'm struggling to understand the notations of the p.d.f. and c.d.f. I read in a book. I understand the following line:
P(a < X < b) = $\int_a^b$ f(x)dx (1)
But the corresponding distribution function of f(x) is shown to be:
F(x) = P(X $\le$ x) = $\int_{-\infty}^x$ f(y)dy (2)
Why is it f(y)dy and not f(x)dx? Isn't this just an extension of (1) to reduce the lower limit from a to -$\infty$ and change the upper limit from b to some value x? To me this implies integrating some completely different function involving y variables rather than the p.d.f. involving x.
Likewise the bivariate c.d.f. is given as:
F(x,y) = P(X < x, Y < y) = $\int_{-\infty}^y\int_{-\infty}^x$ f(u,v)dudv (3)
Again, why not f(x,y)dxdy?
Many thanks

The names are irrelevant, as long as different things are represented by different names. $x$ is already used as the variable of $F(x)$, so you cannot use it again as the integrating variable.
