could someone please tell me how I can prove/confirm the information on this page? specifically the part about calculating the mean, I have my doubts about the -b.
Many thanks
could someone please tell me how I can prove/confirm the information on this page? specifically the part about calculating the mean, I have my doubts about the -b.
Many thanks
You are right, there is a bad typo. It should be $\dfrac{1}{\lambda}+b$. The easiest way to see it is that the density function is the usual one, shifted to the right by an amount $b$. That shifts the mean of the "ordinary" exponential to the right by $b$.
We could also integrate. There is no real point, since the geometric argument above is enough. But for completeness, the mean is $\int_b^\infty \lambda x e^{-\lambda (x-b)}dx.$ To integrate, make the change of variable $u=x-b$. We get $\int_0^\infty (u+b)\lambda e^{-\lambda u}du.$ Break this up into the sum $\int_0^\infty u\lambda e^{-\lambda u}du +\int_0^\infty b\lambda e^{-\lambda u}du.$ The first integral is the familiar mean $\dfrac{1}{\lambda}$ of the standard exponential. The second is simply $b$.