I am trying to solve this problem:
A motorist just had an accident. The accident is minor with probability $0.75$ and is otherwise major. Let $b$ be a positive constant. If the accident is minor, then the loss amount follows a uniform distribution on the interval $[0,b]$. If the accident is major, then the loss amount follows a uniform distribution on the interval $[b,3b]$. The median loss amount due to this accident is $672$. Calculate $b$.
From my point of view, we knew that $$ P(X \leq b) = 0.75 $$ Hence, the median is lies in $[0, b]$. The cumulative function for values in this range is $$\frac{x}{b}$$ Then, to find $b$, we just need to solve $$ \frac{x}{b} = 0.5.$$
However, as the answer given, to find the value of $b$, we need to solve $$ P(X \leq x \mid x < b) = 0.5, $$ which is $$ \frac{1}{0.75b} = 0.5. $$
May I know why do we need this conditional probability? Thank you.
If $X$ is the amount of the loss, then $F(x)$ should only reach 0.75 at b, not 1. The median is the value of x where $F(x) = 0.5$. You are correct that it climbs linearly from 0 at X=0. The area is $\frac{bh}{2} = 0.5$. The slope is $\frac{0.75}{b}$, so at x, the height is $\frac{0.75x}{b}$, giving $$\frac{1}{2}x\cdot\frac{0.75x}{b} = 0.5 $$ The median x is given as 672. Solve for b.