Sorry if this is a silly question, we just started covering pdf/cdf for continuous random variables. For example, if you had a CDF given by
$$
F(x)=\begin{cases}
0&,\quad x<0\\
b&,\quad 0\leq x<1\\
c&, \quad 1\leq x <2\\
1&, \quad x\geq 2
\end{cases}
$$
for some constants $b$ and $c$. We know that the PDF is $\frac{d}{dx}F(x)$, so does this mean that the equation for the PDF is $f(x)=0 $? This doesn't really make sense to me because the probability for values of this random variable $X$ is the area under the curve of the pdf, and the area is going to be $0$ for any limits of integration if this was the case. I suppose I can tell that something is wrong because $\int^\infty_{-\infty}f(x)\neq 1$ if $f(x)=0$.
Best Answer
$F$ is the cdf of a discrete random variable. Discrete random variables have no pdf's.