Missing $\lambda$ in Cauchy distribution

Question

I have to following equality in my lecture notes

$$\frac{1}{\pi \lambda} \int_{-\infty}^x \mathrm{arctan}'\left(\frac{y-\mu}{\lambda}\right)dy = \frac{1}{\pi} \int_{-\infty}^{\frac{x-\mu}{\lambda}} \mathrm{arctan}' \left(y\right)dy $$

and I wonder where the outer $\lambda$ went?

Answer 1

Change of variables: substituting $\lambda z=y-\mu$ implies that $dz = \lambda \, dx$, which cancels the outer $\lambda$. Which you would hope for, since you want the limit of the distribution function as $x \to \infty$ to be independent of $\lambda$.