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Definition/Summary
The exponential distribution is a probability distribution that describes a machine that it equally likely to fail at any given time.
Equations
f(t) = e^{-\lambda t} \lambda
Extended explanation
A machine is equally likely to fail at any given time. For any t, the probability of failure in the interval (t, t + dt) is \lambda dt. So the probability that it doesn't fail in that interval must be 1 - \lambda dt
Let us calculate the probability that it doesn't fail in the interval (0,t). Divide t into n equal parts. Each part then has size \frac{t}{n}. The probabilities that it doesn't fail in the intervals (0,\frac{t}{n}), (0,2\frac{t}{n}), ... are
1 - \lambda \frac{t}{n}
(1 - \lambda \frac{t}{n})^2 ,
... respectively. Therefore we find that the probability that it doesn't fail in the interval (0,t) is approximately
(1 - \lambda \frac{t}{n})^n .
The exact answer is the limit of the above expression as n \rightarrow \infty, i.e.
e^{-\lambda t}.
We can now use this to find the probability density function f(t) that it fails for the first time in the interval (t,t+dt). Clearly, this is the probability that it doesn't fail in the interval (0,t) times the probability that it fails in the interval (t,t+dt).
f(t) = e^{-\lambda t} \lambda.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
The exponential distribution is a probability distribution that describes a machine that it equally likely to fail at any given time.
Equations
f(t) = e^{-\lambda t} \lambda
Extended explanation
A machine is equally likely to fail at any given time. For any t, the probability of failure in the interval (t, t + dt) is \lambda dt. So the probability that it doesn't fail in that interval must be 1 - \lambda dt
Let us calculate the probability that it doesn't fail in the interval (0,t). Divide t into n equal parts. Each part then has size \frac{t}{n}. The probabilities that it doesn't fail in the intervals (0,\frac{t}{n}), (0,2\frac{t}{n}), ... are
1 - \lambda \frac{t}{n}
(1 - \lambda \frac{t}{n})^2 ,
... respectively. Therefore we find that the probability that it doesn't fail in the interval (0,t) is approximately
(1 - \lambda \frac{t}{n})^n .
The exact answer is the limit of the above expression as n \rightarrow \infty, i.e.
e^{-\lambda t}.
We can now use this to find the probability density function f(t) that it fails for the first time in the interval (t,t+dt). Clearly, this is the probability that it doesn't fail in the interval (0,t) times the probability that it fails in the interval (t,t+dt).
f(t) = e^{-\lambda t} \lambda.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!