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:

\(\ds \map {I_\Pr} {X_n}\)

\(=\)

\(\ds \sum_{i \mathop = 1}^n x_i \map \Pr {E_i}\)

Definition of Integral of Positive Simple Function

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop = 1}^n x_i \map \Pr {X = x}\)

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:

\(\ds \map {X_{n + 1} } \omega - \map {X_n} \omega\)

\(=\)

\(\ds \sum_{i \mathop = 1}^{n + 1} x_i \map {\chi_{E_i} } \omega - \sum_{i \mathop = 1}^n x_i \map {\chi_{E_i} } \omega\)

\(\ds \)

\(=\)

\(\ds x_{n + 1} \map {\chi_{E_{n + 1} } } \omega\)

\(\ds \)

\(\ge\)

\(\ds 0\)

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:

\(\ds \int X \rd \Pr\)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \int X_n \rd \Pr\)

\(\ds \)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \sum_{i \mathop = 1}^n x_i \map \Pr {X = x}\)

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop = 1}^\infty x_i \map \Pr {X = x}\)

Definition of Infinite Series

$\blacksquare$