This is part of a short series on the common life data distributions.
The Exponential distribution is popular and useful in isolated situations. It has some nice features and flexibility that support it’s popularity. This short article focuses on 7 formulas of the Exponential Distribution.
If you want to know more about fitting a set of data to a distribution, well that is in another article.
It has the essential formulas that you may find useful when answering specific questions. Knowing a distribution’s set of parameters does provide, along with the right formulas, a quick means to answer a wide range of reliability related questions.
Parameters
The 1-parameter Exponential distribution has a scale parameter.
The scale parameter is denoted here as lambda (λ). It is equal to the hazard rate and is constant over time. Be certain to verify the hazard rate is constant over time else this distribution may lead to very poor results and decisions.
The inverse of the scale parameter 1 / λ is commonly denoted by theta, θ, and called the MTBF or MTTF.
Probability Density Function (PDF)
When t ≥ 0 then the probability density function formula is:
$$ \displaystyle\large f\left( t \right)=\lambda {{e}^{-\lambda t}}$$
A plot of the PDF provides a histogram-like view of the time-to-failure data.
Cumulative Density Function (CDF)
F(t) is the cumulative probability of failure from time zero till time t. Very handy when estimating the proportion of units that will fail over a warranty period, for example.
$$ \displaystyle\large F(t)=1-{{e}^{-\lambda t}}$$
Reliability Function
R(t) is the chance of survival from time zero till time t. Instead of looking for the proportion that will fail the reliability function determine the proportion that is expected to survive.
$$ \displaystyle\large R(t)={{e}^{-\lambda t}}$$
Conditional Survivor Function
The m(x) function provides a means to estimate the chance of survival for a duration beyond some known time, t, over which the item(s) have already survived. What is the probability of surviving time x given the item has already survived over time t?
$$ \displaystyle\large m(x)={{e}^{-\lambda x}}$$
Note the exponential distribution is memoryless, meaning the probability of failure over any given duration is not affected by how long the item has been in use. Since the hazard rate is constant for any duration, the chance of failure the first hours is the same as the 100th or 1,000th hour.
Mean Residual Life
This is the cumulative expected life over time x given survival till time t.
$$ \displaystyle\large u(t)=\frac{1}{\lambda }$$
Hazard Rate
This is the instantaneous probability of failure per unit time.
$$ \displaystyle\large h(t)=\lambda $$
That is pretty simple, which makes the integration for the the cumulative hazard rate simple as well.
Cumulative Hazard Rate
This is the cumulative failure rate from time zero till time t, or the area under the curve described by the hazard rate, h(x).
$$ \displaystyle\large H(t)=\lambda t$$
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