The slide explains the discrete case of the probability distribution of the sum of two random variables, $Z = X + Y$. Here's a simplified explanation:

1. General Case (Dependent or Independent Variables):

   • To calculate the probability that $Z = k$, you add up the probabilities of all pairs of values $(x_i, y_j)$ such that $x_i + y_j = k$.
   • Formula:
     $\mathbb{P}(Z = k) = \sum_{i + j = k} \mathbb{P}(X = x_i, Y = y_j).$

2. If $X$ and $Y$ Are Independent:

   • The joint probability $\mathbb{P}(X = x_i, Y = y_j)$ becomes the product of their individual probabilities:
     $\mathbb{P}(X = x_i, Y = y_j) = \mathbb{P}(X = x_i) \cdot \mathbb{P}(Y = y_j).$
   • So, the formula for $Z = k$ simplifies to:
     $\mathbb{P}(Z = k) = \sum_{i + j = k} \mathbb{P}(X = x_i) \cdot \mathbb{P}(Y = y_j).$

Summary:

• Dependent Variables: Use joint probabilities.
• Independent Variables: Use the product of individual probabilities.

Would you like a worked-out example or further clarification? Here are related questions to explore:

1. How do you compute probabilities for continuous random variables?
2. Can this method extend to more than two variables (e.g., $X + Y + W$)?
3. What are the key differences between marginal, joint, and conditional probabilities?
4. How does this work for random variables with non-uniform distributions?
5. What is the role of convolution in summing independent random variables?

Tip: If $X$ and $Y$ are independent and have simple distributions (e.g., uniform or binomial), leveraging their independence can drastically simplify computations.