Let $Y$ be a random variable taking values in $\mathbb{N}$ almost surely, and which admits an expectation. Show that $$\mathbb{E}(Y) = \sum_{k=1}^{+\infty} \mathbb{P}(Y \geqslant k)$$
Let $Y$ be a random variable taking values in $\mathbb{N}$ almost surely, and which admits an expectation. Show that
$$\mathbb{E}(Y) = \sum_{k=1}^{+\infty} \mathbb{P}(Y \geqslant k)$$