What is the difference between the intuition behind Gamma and Weibull distributions? Is there any relationship between the two densities?
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Both the gamma and Weibull distributions can be seen as generalisations of the exponential distribution. If we look at the exponential distribution as describing the waiting time of a Poisson process (the time we have to wait until an event happens, if that event is equally likely to occur in any time interval), then the $\Gamma(k, \theta)$ distribution describes the time we have to wait for $k$ independent events to occur.
On the other hand, the Weibull distribution effectively describes the time we have to wait for one event to occur, if that event becomes more or less likely with time. Here the $k$ parameter describes how quickly the probability ramps up (proportional to $t^{k-1}$).
We can see the difference in effect by looking at the pdfs of the two distributions. Ignoring all the normalising constants:
$$ f_{\Gamma}(x) \propto x^{k-1} \exp\left(-\frac{x}{\theta}\right) \\ f_{W}(x) \propto x^{k-1} \exp\left(-\left(\frac{x}{\lambda}\right)^k\right) $$
As you can see from this, the pdf for the Weibull distribution drops off much more quickly (for $k > 1$) or slowly (for $k < 1$) than the gamma distribution. In the case where $k = 1$, they both reduce to the exponential distribution.
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3$\begingroup$ This is very helpful! Naturally both distributions are often used for variables other than waiting time, so that the derivation and motivation may be very different. On a different note, Weibull and Poisson deserve their initial capitals because they are named after people, but many (I'd venture most) discussions of exponential and gamma don't use initial capitals. $\endgroup$– Nick CoxCommented Nov 25, 2013 at 12:07
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$\begingroup$ Fixed the capitalisation - I was tired when I wrote the answer, and the OP had capitalised Gamma, so I ran with it. You are of course right that not every use of these distributions is a waiting time, but I think this provides the best intuition. If there's another good way of thinking about it, I'd love to hear it. $\endgroup$ Commented Nov 25, 2013 at 14:48
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$\begingroup$ I don't have a better general story! Capitalisation (e.g. gamma/Gamma) is naturally a matter of convention when surnames are not involved. $\endgroup$– Nick CoxCommented Nov 25, 2013 at 16:30
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$\begingroup$ What do you mean by "if that event becomes more or less likely with time"? $\endgroup$ Commented Oct 26, 2023 at 15:21
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$\begingroup$ I see I upvoted this years ago. Upon re-examining it today, I would like to suggest re-writing $f_W$ in the much more revealing form $$f_W(x)\ \propto\ \left(\frac{x}{\lambda}\right)^k\,\exp\left(-\left(\frac{x}{\lambda}\right)^k\right)\,\frac{\mathrm d(x/\lambda)^k}{(x/\lambda)^k},$$ thereby impressing upon the reader that the relationship is simply one of re-expressing $x^k$ as a variable $u.$ $\endgroup$– whuber ♦Commented Jan 31 at 16:34