Lifetime of a component

Problem

The mean lifetime of a component is $1000$ hours (exponential distribution). What is the probability that it lasts more than $1500$ hours?

Solution

The mean of an exponential is $1/\lambda$, so $\lambda = 1/1000 \ \text{ore}^{-1}$. The probability of surviving beyond time $t$ is $P(X>t) = 1 - F(t) = e^{-\lambda t}.$ With $t=1500$: $P(X>1500) = e^{-1500/1000} = e^{-1{,}5} \approx \boxed{0{,}223}.$ About $22\%$ of the components exceed $1500$ hours.

Links

Topics: Distribuzioni probabilita
Concepts: Distribuzione esponenziale · Funzione di ripartizione
Skills: Calcolo probabilita · Modellizzare
Exercise type: Problema probabilita