This is part of a short series on the common life data distributions.
The Gamma distribution is routinely used to describe systems undergoing sequences of events or shocks which lead to eventual failure. Also used to describe renewal processes. This short article focuses on 7 formulas of the Gamma 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 scale parameter, λ which is greater than zero represents the rate or frequency of events or shocks.
The shape parameter, k may be interpreted as the number of shocks till failure (k as an integer only – and the distribution then is also called an Erlang distribution) or as a measure of the ability to resist shocks when not limited to being an integer.
The shape parameter, k, has the following characteristics:
When k < 1, then f(0) = ∞ and there is no mode.
When k = 1, then f(0) = λ and the gamma distribution reduces to an exponential distribution with failure rate λ.
When k >1 then f(0) = 0.
With a large k value the distribution approaches a normal distribution with
$$ \displaystyle\large \mu =\frac{k}{\lambda },\sigma =\sqrt{\frac{k}{{{\lambda }^{2}}}}$$
The gamma distribution incorporates the use of the complete or incomplete gamma function, Γ(k) or γ(k,t), respectively.
Probability Density Function (PDF)
When t ≥ 0 then the probability density function formula is:
$$ \displaystyle\large f(t)=\frac{{{\lambda }^{k}}{{t}^{k-1}}}{\Gamma \left( k \right)}{{e}^{-\lambda t}}$$
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.
$$ \displaystyle\large \begin{array}{l}F\left( t \right)=1-{{e}^{-\lambda t}}\sum\limits_{n=0}^{k-1}{\frac{{{\left( \lambda t \right)}^{n}}}{n!}}\\F\left( t \right)=\frac{\gamma \left( k,\lambda t \right)}{\Gamma \left( t \right)}\end{array}$$
Reliability Function
R(t) is the chance of survival from from time zero till time t. Instead of looking for the proportion that will fail the reliability function determine the proportion that are expected to survive.
$$ \displaystyle\large \begin{array}{l}R\left( t \right)={{e}^{-\lambda t}}\sum\limits_{n=0}^{k-1}{\frac{{{\left( \lambda t \right)}^{n}}}{n!}}\\F\left( t \right)=\frac{\Gamma \left( k,\lambda t \right)}{\Gamma \left( t \right)}\end{array}$$
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 the probability of surviving time x given the item has already survived over time t?
$$ \displaystyle\large m\left( t \right)=\frac{R\left( t+x \right)}{R\left( t \right)}=\frac{\Gamma \left( k,\lambda t+\lambda x \right)}{\Gamma \left( k,\lambda t \right)}$$
Mean Residual Life
This is the cumulative expected life over time x given survival till time t.
$$ \displaystyle\large u\left( t \right)=\frac{\int_{t}^{\infty }{\Gamma \left( k,\lambda x \right)dx}}{\Gamma \left( k,\lambda t \right)}$$
Hazard Rate
This is the instantaneous probability of failure per unit time.
$$ \displaystyle\large \begin{array}{l}h\left( t \right)=\frac{{{\lambda }^{k}}{{t}^{k-1}}}{\Gamma \left( k \right)\sum\limits_{n=0}^{k-1}{\frac{{{\left( \lambda t \right)}^{n}}}{n!}}}\\h\left( t \right)=\frac{{{\lambda }^{k}}{{t}^{k-1}}}{\Gamma \left( k,\lambda t \right)}{{e}^{-\lambda t}}\end{array}$$
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).
$$ \displaystyle\large \begin{array}{l}H\left( t \right)=\lambda t-\ln \left[ \sum\limits_{n=0}^{k-1}{\frac{{{\left( \lambda t \right)}^{n}}}{n!}} \right]\\H\left( t \right)=-\ln \left[ \frac{\Gamma \left( k,\lambda t \right)}{\Gamma \left( k \right)} \right]\end{array}$$
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