I have to calculate the living span of a chip from the given cdf:
$F(x) = \begin{cases}0 & \quad x\leq 0 \\ 1-exp(-\beta x^\alpha) & \quad x>0\end{cases}$
with $\alpha > 0$ and $\beta > 0$, and $x$ being the lifespan in years.
Part of the exercise is to see how large the median of the life span is for general $\alpha$ and $\beta$. I know, that to find year of the median I set $F(x)=0.5$, but what do they mean by a general $\alpha$ and $\beta$?
Just in case it might be needed, though I can't see that it should. First part was to find the pdf of the cdf. Took the derivative of $1-exp(-\beta x^\alpha)$. Second part we had a given $\alpha = 2$ and $\beta = 1/4$ and had to find out the probability if the chip survived over 4 years, and what the probability was that the chip stops working in the time interval [5;10]. Calculated the first part by saying $1-F(4)$ and the second part $F(10)-F(5)$.
Best Answer
You did everything correctly so I will just answer the sentence with a question mark in your question:
"For general $\alpha$ and $\beta$" means that you should write your answer as a function of $\alpha$ and $\beta$. The median life span $x_0$ is
$$0.5=1-\exp\left(-\beta x_0^\alpha\right) \implies x_0=\left(\frac{1}{\beta}\ln 2\right)^{\frac{1}{\alpha}}.$$