Let $X$ be a continuous random variable with PDF
$$f_X(x)=
\begin{cases} cx(1-x),
& \text{$0<x<1$} \\ 0,
& \text{elsewhere}
\end{cases}$$
Find the median of $X$.
My question is how I am only given PDF, to calculate median, do I need to find CDF for it? And how?
Best Answer
You do not need to calculate anything.
The distribution is symmetric about $x=\frac 12$. (Draw it.)
So, median and mean are the same and are equal to $\frac 12$.