Expectation of Real-Valued Discrete Random Variable/Lemma

Lemma
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ be a discrete real-valued random variable such that:


 * $\map X \omega \ge 0$ for all $\omega \in \Omega$.

Then:


 * $\ds \int X \rd \Pr = \sum_{x \in \Img X} x \map \Pr {X = x}$

Proof
Since $X$ is a discrete random variable, there exists a sequence $\sequence {x_i}_{i \in \N}$ of distinct real numbers such that:


 * $\Img X = \set {x_i : i \in \N}$

For each $i$, let:


 * $E_i = \set {X = x_i}$

Then, we can write:


 * $\ds \map X \omega = \sum_{i \mathop = 1}^\infty x_i \map {\chi_{E_i} } \omega$

for each $\omega \in \Omega$.

Since $X$ is $\Sigma$-measurable, we have:


 * $E_i$ is $\Sigma$-measurable for each $i$

from Measurable Functions Determine Measurable Sets.

For each $n \in \N$, define $X_n : \Omega \to \R$ by:


 * $\ds \map {X_n} \omega = \sum_{i \mathop = 1}^n x_i \map {\chi_{E_i} } \omega$

for each $\omega \in \Omega$.

Then $X_n$ is a positive simple function for each $n$.

So, we have:

From Integral of Positive Measurable Function Extends Integral of Positive Simple Function, we have:


 * $\ds \map {I_\Pr} {X_n} = \int X_n \rd \Pr$

That is:


 * $\ds \int X_n \rd \Pr = \sum_{i \mathop = 1}^n x_i \map \Pr {X = x}$

We aim to apply the Monotone Convergence Theorem for Positive Simple Functions.

We have, from the definition of infinite series:


 * $\ds \map X \omega = \lim_{n \mathop \to \infty} \sum_{i \mathop = 1}^n x_i \map {\chi_{E_i} } \omega = \lim_{n \mathop \to \infty} \map {X_n} \omega$

for each $\omega \in \Omega$, so:


 * $X_n \to X$ pointwise.

Finally, for each $n \in \N$, we have:

for each $\omega \in \Omega$.

So:


 * $\map {X_{n + 1} } \omega \ge \map {X_n} \omega$ for each $n \in \N$ and $\omega \in \Omega$.

So, from Monotone Convergence Theorem for Positive Simple Functions, we have: