$$ f(x) = \begin{cases} 0 & \text{if $x < 0$} \\ x^2 & \text{if $0 ≤ x < \mu$} \\ α + βx & \text{if $\mu ≤ x ≤ 10$} \\ 0 & \text{if $x ≥ 10$} \end{cases} $$
Considering $\Pr(x < µ) = 1/3$. Questions:
Find $\mu$.
Find $\alpha$.
Find $\beta$.
Find the expected value $E(x)$.
I was solving all questions related to probability density functions with not much difficulty, but this one is driving me nuts. What I have done so far:
$$ \int_{0}^{\mu} x^2 \, dx + \int_{\mu}^{10} (\alpha + \beta x) \, dx = 1 $$
which would lead to something like:
$$ (\frac{x^3}{3} \bigg|_{0} ^{\mu}) + (\alpha \cdot x+ \beta \cdot \frac{x^2}{2}\bigg|_{\mu}^{10}) = 1 $$
Anyone know how to proceed?