Show that the expected value of $X$ is equal to $E(x)$

Question

Consider a random variable $X : Ω → \{1, . . . , 15\}$, that takes on integer values in the set $\{1, . . . , 15\}$. Show that the expected value of $X$ is equal to: $$E(X) = \sum_{i=1}^{15} P(X ≥ i)$$

Answer 1

\begin{eqnarray} E(X)&=& \sum_{k=1}^{15}k \cdot \mathbb P\{X=k\}\\ &=& \color{red}{\mathbb P\{X=1\}}\\ &+& \color{red}{\mathbb P\{X=2\}}+ \color{blue}{\mathbb P\{X=2\}}\\ &+& \color{red}{\mathbb P\{X=3\}}+ \color{blue}{\mathbb P\{X=3\}}+ \color{green}{\mathbb P\{X=3\}}\\ & \vdots &\\ &+& \color{red}{\mathbb P\{X=15\}}+ \color{blue}{\mathbb P\{X=15\}}+ \color{green}{\mathbb P\{X=15\}}+\dots + \mathbb P\{X=15\}\\ &=& \color{red}{\mathbb P\{X\geq 1\}}+ \color{blue}{\mathbb P\{X\geq 2\}}+ \color{green}{\mathbb P\{X\geq 3\}}+\dots + \mathbb P\{X=15\}\\ &=& \sum_{k=1}^{15} \mathbb P\{X\geq k\}\\ \end{eqnarray}

Answer 2

Hint

$$\mathbb P\{X\geq i\}=\sum_{k=i}^{15}\mathbb P\{X=k\}.$$