I am strangely stuck on how to do this probability problem on certain sections upon going through some past papers:
part a)
For this part, do I compute two integrals from $[1,2]$ using $f(x)=3a$ and $[2,3]$ using $f(x)=a(x-5)(1-x)?$
I am kind of stuck on c), if they gave me one pdf $f(x)$ I think I know how to do this, but when they give me a pdf $f(x)$broken down with respect to a given domain, I am confused on how to find the mean and median. Can anyone show me how to do this, or give me some hints? I would appreciate it.
For part c, you are given that the value of $b$ is 5. Using this information, and the fact that $\int_{x=0}^{5}{f(x) dx} = 1$, you can compute the value of $a$. This would define $f(x)$ completely.
The median $m$ is defined as the value in the domain of the random variable $X$ such that $Pr(X\leq m) = \frac{1}{2}$ and $Pr(X\geq m) = \frac{1}{2}$. As pointed out by @WaveX in the comments, you simply need to compute $m$ such that $\int_{0}^{m} f(x) = \frac{1}{2}$.
As for the mean, the mean of a probability distribution is simply the expected value of that variable.
Hope this clears up your question.