Cumulative distribution function derivation

Question

Can someone please explain me why in the first example limits of integration are from - infinity to s and then from 1 to s but in the second c.d.f the limits of the integration are from - a to a and -1 to s/a ?

Thank you!

Answer 1

The definition of the cumulative distribution function (corresponding to a density function $f$) is always $F(s) = \int_{-\infty}^s f(x)\,dx$, with lower limit $-\infty$ and upper limit $s$. This choice will always be correct.

If a function $f(x)$ is identically zero for all $x<c$ for some constant $c$, then $\int_{-\infty}^s f(x)\,dx = \int_c^s f(x)\,dx$ (just from how integrals are defined). That's why in the first example we can replace $\int_{-\infty}^s f_1(x)\,dx$ with $\int_1^s f_1(x)\,dx$, and why in the second example we can replace $\int_{-\infty}^s f_2(x)\,dx$ with $\int_{-a}^s f_2(x)\,dx$.

Finally, the fact that $\int_{-a}^s \frac2{\pi a^2}\sqrt{a^2-x^2}\,dx = \int_{-1}^{s/a} \frac2{\pi}\sqrt{1-y^2}\,dy$ is a fact from calculus, due to the change of variables $y=x/a$; it doesn't have anything to do with the definition of the cumulative density function, but just provides one person's way of evaluating this particular cumulative density function.