Weibull Distribution v/s Gamma Distribution What is the difference between the intuition behind Gamma and Weibull distributions? Is there any relationship between the two densities ?
Kindly help.
 A: 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.
