I have an exponential distribution modelling the lifetime of certain product:
$$f(x)=\left\{\begin{array}{l}\lambda e^{-\lambda x} &,\text{ if } x\geq 0\\0 &, \text{ if } x < 0\end{array}\right.$$
A text has the following statement: In this case, $\lambda$ is the average lifetime and $x$ is a failure time.
I'd like to know why the average lifetime is not the expected value, ie, $1/\lambda$.
Thank you.