I need to compute a CDF knowing the PDF of a random variable, so nothing fancy.
I find the constant c by stating that:
I find $c = 0.5114$
But when I want to determine the CDF and its bounds, I have a problem
My CDF becomes negative, and for a non-negative function, it is quiet a problem Do you see the problem and could you give me a hint ? Thanks in advance !
The CDF is an antiderivative of the PDF but generally not the "trivial" one given to us by the elementary calculus rules. To illustrate this, instead of doing your problem, I will do a related problem, namely finding the CDF of a random variable with PDF $f(x)=\begin{cases} 0 & x<0 \\ e^{-x} & x \geq 0 \end{cases}$. Then the CDF is
$$F(x)=\int_{-\infty}^x f(y) dy=\begin{cases} 0 & x<0 \\ \int_0^x e^{-y} dy & x \geq 0 \end{cases}.$$
We compute this integral:
$$\int_0^x e^{-y} dy = -e^{-x}-(-1)=1-e^{-x}.$$
Note the term from the endpoint $0$, which is necessary in order to keep $F$ from being negative, since the "trivial" antiderivative is indeed negative.
Essentially the same thing happens in your problem: there will be a term from the endpoint $1$ which keeps $F$ from being negative.