I want to make sure that I solved the next problem correctly:
What is the probability that a randomly chosen $x$ in $[0,+\infty)$ is less than 1, if the probability density function on $[0,+\infty)$ is given by: $ρ(x)=\frac{r}{1+x^2}$ for some $r$.
This is what I did:
I make sure the PDF met the requirements, that is: $ρ(x) \geq 0$ and $\int_{D}ρ(x)dx = 1$ Therefore, I solved: $\int_0^{+\infty} \frac{r}{1+x^2}dx = r\int_0^{+\infty} \frac{1}{1+x^2}dx = \frac{\pi r}{2}$, therefore $r$ must equal $\frac{2}{\pi}$ for $ρ(x)$ to be a PDF.
I found the probability by integrating the new PDF over the domain $[0,1]$ $\int_0^1 \frac{1}{1+x^2}\frac{2}{\pi}dx = \frac{1}{2}$
I just want to make sure that my result is correct, any help would be appreciated :D Thanks in advance!