Mean Value of Weibull distribution for small Beta

[Vladimir Balabanov]

Hello,
I am a software engineer. Our company wants to present to the user the opportunity to visualize couple of common probability distributions (normal, lognormal, uniform, Weibull) and corresponding general characteristics (mean, standard deviation). I have a really basic question that I can not quite figure out.
Most likely I am missing some really basic
thing.

By looking at the formula for the mean value of the Weibull distribution it seems that the smaller the value of Beta the larger the mean value. Particularly for the case of Beta < 1.
(The other 2 parameters are constant)

However, looking at the PDF plots in the section "Characteristics of the Weibull distribution" on your site it seems that the mean value should get closer to 0 as Beta decreases from 1 to 0. (PDF sort of leans to "y" axis).

Probably I am not interpreting PDF plot quite right...

Could you, please, clarify my confusion?
Please, e-mail me if it is more convenient.

Thank you very much in advance for your help.

Vladimir

[Adam]

Actually this is one of the reasons why the mean is such a misleading metric. As you pointed out the mean increases as beta decreases (for the same eta and for 0<beta<1). However, if you calculate the probability at the mean you'll see that there is a higher probability of occurance at the lower beta. In reliability engineering this means that a higher portion of the population will fail by the mean life. It also means that a higher percentage will fail initially, but there is also a small percentage that will survive a long period of time (compared to the initial failures). This portion is what is pushing the mean to the right. In other words you have a much higher spread (or standard deviation). The smaller the beta the higher the spread. Relating to the pdf, even though the curve leans to y-axis as beta goes to zero, you'll notice that it also extends further on the x-axis (i.e., approaches zero less rapidly).