Tail Sum¶
If \(\mathbf{X}\) is a non-negative random variable, then:
-
For discrete:
\[ E(\mathbf{X}) = \sum_{k=1}^n P(\mathbf{X}=k) \] -
For continuous:
\[ E(\mathbf{X}) = \int_0^{\infty} P(\mathbf{X}>k) \, dt \]
Example :
\[
E(\mathbf{X}) = \sum_{k=1}^n P(\mathbf{X}=k) = \sum_{k=0}^n k P(\mathbf{X}=k)
\]
\[
\begin{matrix}
P(\mathbf{X} \ge 1) & P_1 + P_2 + P_3 + \cdots + P_{n-1} + P_n \\
P(\mathbf{X} \ge 2) & \quad + P_2 + P_3 + \cdots + P_{n-1} + P_n \\
P(\mathbf{X} \ge 3) & \quad \quad + P_3 + \cdots + P_{n-1} + P_n \\
\vdots & \vdots \\
P(\mathbf{X} \ge n-1) & \quad \quad \quad \quad + P_{n-1} + P_n \\
P(\mathbf{X} \ge n) & \quad \quad \quad \quad \quad + P_n \\
\end{matrix}
\]
\[
P(\mathbf{X} \ge k) = 1 - P(\mathbf{X} \leq k)
\]