Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$
$$T=\sum_{k=1}^\infty\mathbf 1_{T\geqslant k}=\sum_{k=1}^\infty k\cdot \mathbf 1_{T=k}$$
Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$
$$T=\sum_{k=1}^\infty\mathbf 1_{T\geqslant k}=\sum_{k=1}^\infty k\cdot \mathbf 1_{T=k}$$