Suppose I'm given a CDF: $Cf(x)$ for $x<0$ and 1 for $x\ge 0$. I need to find $C$. What is my strategy? I don't know calculus very well, unfortunately. So far I only have the following solution:
- Find the derivative
- Take an integral with the limits from $-\infty$ to $0$ and make it equal $1$.
- Find $C$
But I am not sure whether I'm right because it's weird to take a derivative and put it into an integral assuming that I already know the antiderivative. As you see, I'm totally confused.
For cumulative distribution function $F$, we have
$$\lim_{x\rightarrow \infty}F(x)=1$$
So indeed, if the variable is absolutely continuous, then
$$\int_\mathbb R f(x)dx=1,$$
where $f$ is the derivative of $F$. Hence if you have $f_2=Cf$, then
$$\int_\mathbb R f_2(x)dx=C \Rightarrow f=\frac{f_2}{C}.$$
In this case we have $$F(0)=1\Leftrightarrow CF(0)=C,$$
so all you need to do is evaluate $CF(0)$, which is possibly
$$CF(0)=C\int_\mathbb {R_{\leq 0}} f(x)dx$$.