This slide discusses properties of expectation in probability theory and provides a formal definition and theorem. Here's the explanation in English:

Definition (Almost Surely - "Presque Sûrement (ps)")

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, and let $\mathcal{P}$ be a property that may or may not hold for all $\omega \in \Omega$.
We say that $\mathcal{P}$ is true almost surely (ps) if there exists a set $A \in \mathcal{A}$ such that:

• $\mathbb{P}(A) = 0$ (the probability of $A$ is zero),
• and the property $\mathcal{P}$ holds for all $\omega \notin A$.

This means the property $\mathcal{P}$ holds everywhere except possibly on a set of probability zero.

Theorem (Properties of Expectation)

Let $X$ and $Y$ be two discrete random variables defined on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$. The following properties hold:

1. Finiteness of Expectation:

   • $\mathbb{E}[X]$ is finite if and only if $\mathbb{E}[|X|]$ is finite.
     (The expectation of $X$ converges if and only if the expectation of its absolute value converges.)

2. Comparison Property:

   • If $|X| \leq Y$ and $\mathbb{E}[Y]$ is finite, then $\mathbb{E}[X]$ is finite as well.
     (Bounded random variables by integrable variables ensure their expectation is finite.)

3. Bounding Expectation:

   • If $a \leq X \leq b$ almost surely, then $a \leq \mathbb{E}[X] \leq b$.
     (The expectation is bounded by the bounds of the random variable.)

4. Almost Sure Property:

   • If $X = a$ almost surely (ps), then $\mathbb{E}[X] = a$.
     (If a random variable is constant almost surely, its expectation is the same constant.)

5. Absolute Expectation:

   • $|\mathbb{E}[X]| \leq \mathbb{E}[|X|]$.
     (The absolute value of expectation is always less than or equal to the expectation of the absolute value.)

Would you like to explore this further or see related examples? Here are some possible follow-up questions:

1. What is the significance of "almost surely" in probability theory?
2. Can you provide an example of bounded random variables and their expectations?
3. How is the property $|\mathbb{E}[X]| \leq \mathbb{E}[|X|]$ used in applications?
4. Could you explain how to prove one of these properties, such as $|\mathbb{E}[X]| \leq \mathbb{E}[|X|]$?
5. What happens to expectations when $X$ is unbounded?

Tip: Always ensure finiteness of expectations when working with integrals or series, as this ensures the convergence of probability calculations.