Density function and constant k

Question

I was trying to solve a problem, but I got stuck. Suppose that a given demand function follows that P = 100 - Q (where P is price and Q is quantity), and the problem says that the variable Q has a certain density function such that:

How can I find that density function?

Thanks!

Answer 1

PDF of $Q$ is $r-q$ for some $r, q \in [20,30]$

$\phi(Q) = 100-P$, so $\phi^{-1}(P) = 100-Q$.It's a decreasing function, so $$ P(P \leq p) = P(Q>1-p) $$ Now compute the probabilty that $Q$ is between $100-p$ and $30$, then differentiate to get $f_P(p)$.