1
$\begingroup$

The time to failure (in hours) of a component is a continuous random variable $T$ with the probability density function $$f(t)=\begin{cases}\frac{1}{10}e^{-t/10}~~~t>0\\ 0, ~~~~~~~~~~~~~t\leq 0\end{cases}$$ $10$ of these components are installed in a system and they work independently. Then, the probability that NONE of these fail before $10$ hours is ____?

My try: So $T$ has exponential distribution. So the CDF is $F(T)=1-e^{-t/10}$. So the probability that one of the component will fail before ten hours is $$P(T\leq 10)=F(10)=1-e^{-1}.$$ So taking the complement we get the probability that none of them fail before $10$ hours is $e^{-1}$.

Is my solution correct? Any suggestion and correction will be helpful. Thanks.

  • 2
    $e^{-1}$ is the probability that one component independently of the others will work more than 10 hours, They ask for the probability that all 10 components wil work more than 10 hours. By the independence this probability is $(e^{-1})^{10}$ – 2017-01-11

1 Answers 1

2

Hint:

Let $T_i$ be the time of failure.

You should consider $P(\min_{i=1,\ldots, 10} T_i> 10)$ rather than $P(T_1 > 10)$