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I may be expressing this in completely the wrong fashion, but please bear with me - I'm an engineer.

Suppose I have an electric fuse in my house, and I know its probability of failure over time is distributed with probability density function $f(t) = 1-e^{-t}$ on $[0,\infty]$. However once it's failed, the probability of failure goes to zero because it cannot fail again.

Is it even possible to find $P(fail)$ between $t_1$ and $t_2$? If so, I imagine I need to also find the intersection with the probability that failure did not occur between $0$ and $t_1$?

Any guidance greatly appreciated.

wobytides
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The function $1-e^{-t}$ is not a probability density function.

Let random variable $X$ be the length of life of the fuse. A simple but probably unsatisfactory model for the distribution of $X$ is that $X$ has exponential distribution with parameter $\lambda$. The density function is then $\lambda e^{-\lambda t}$ (for $t\gt 0$). The cumulative distribution function $F(t)$ of $X$, that is, the probability that $X\le t$, is then equal to $1-e^{-\lambda t}$.

It looks as if yours is the case $\lambda=1$, where the mean life $\frac{1}{\lambda}$ is $1$.

The probability of failure between $t_1$ and $t_2$ is $F(t_2)-F(t_1)$. In your case that simplifies to $e^{-t_1}-e^{-t_2}$.

André Nicolas
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  • Ah, sorry, you're completely right about the distribution and the answer makes sense. I was looking at it the completely the wrong way. Thanks a lot for the answer. – wobytides Feb 25 '14 at 21:57
  • You are welcome. You were given the cdf and assumed that (as is more common) you had been given the pdf. – André Nicolas Feb 25 '14 at 22:06