From an electronic component it's known that it lasts $10$ years in
average, and its failure probability is exponentially distributed.
What's the probability that the electronic component lasts only $8$
years or less?
I think I did the maths correctly:
Need to use the distribution function $$F(x)=\begin{cases} 1-e^{-\lambda x} & ,x \geq 0 \\0 & , x < 0\end{cases}$$
From task we have that $\lambda = \frac{1}{10}$. So $$P(X \leq 8) = 1-e^{-\frac{1}{10} \cdot 8} = 0.5507$$
Now I'm not sure about this number, how to put it into an answer what this tells us? I think this number refers to the electronic components that will no longer work because the specifcation from the task refers to components that will fail working ("its failure probability is exponentially distributed").
Is that a correct reason? I would answer it like that: About $55\%$ of electronic components will not work until (including) the 8th year.
I mean I'm having a problem to decide whether the result refers to components that will work, or if it refers to components that will not work. How can I know that?
Best Answer
The statement $P(X\leq8)=0.55$ can be interpreted as a statement concerning "the" electronic component that you intend to buy in a shop tomorrow. It tells you that the probability you can still use it after $8$ years is $0,45$.
Also you can interpret it as a statement over the bunch of all such components: approximately $45%$ percent of them will still be functional after $8$ years.
I hope this helps, but it might be that I did not really understand your question.