In page 21 of *A Problem seminar*, D. J. Newman presents a novel way (at least for me) to determine the expectation of a discrete random variable. He refers to this expression as the **failure probability formula**. His formula goes like this

$f_{0}+f_{1}+f_{2}+\ldots$

where $f_{n}$ is the probability that the experiment fails to produce the desired outcome for $n$ steps.

He goes on to outlining two ways to establish the validity of this formula, but I'm having a hard time to unravel his second proof. I'll insert it here in order for my inquiry to be self-contained:

Since the expected amount of time spent on the $n$ trial is equal to the probability that the $n$ trial occurs, and since this equal to $f_{n-1}$, we do obtain the net expectation of $f_{0}+f_{1}+f_{2}+\ldots$ as asserted.

Can any of you guys explain in greater detail this Newmanian argument? I'd also like to know whether the denomination **failure probability formula** is a standard one in the mainstream of Probability literature.

Thank you very much for your continued support.

alsogiven by the Riemann integral $\int_0^\infty \Pr(X > x) \,dx$, which is the same as what you gave when the values are nonnegative integers. – Michael Hardy Jun 15 '10 at 21:40