17.2. Joint Distribution of Random Variables

When two or more random variables are involved, we use joint distributions:

• Let 𝑋 and 𝑌 be two random variables. Their joint distribution describes the probability structure of the pair.

Joint distributions allow us to understand the relationships between two or more random variables. Specifically, if we have two random variables, X and Y, the joint distribution tells us how likely different combinations of their values are to occur together.

In simpler terms, if we were to plot every occasion when both X and Y take specific values, the joint distribution would provide a complete picture of how these variables interact with each other in terms of probability.

Imagine you are tracking the height (X) and weight (Y) of a group of people. The joint distribution would tell you how often people of specific height and weight combinations occur. For instance, if most people who are 170 cm tall weigh between 60 kg and 80 kg, the joint distribution reflects this relationship, giving you a clearer view of height and weight trends in your dataset.

3.2.1 Joint Probability Mass Function (PMF)

For discrete random variables: 𝑃(𝑋 = 𝑥 ,𝑌 = 𝑦 ) = 𝑝𝑖𝑗𝑖𝑗

The Joint Probability Mass Function (PMF) is essential for discrete random variables. It provides a way to compute the probability of specific outcomes for X and Y simultaneously. The notation P(X = x, Y = y) denotes the probability that random variable X takes the value x while random variable Y takes the value y.

The joint PMF can be visualized as a table, where each cell represents the likelihood of a specific pair of outcomes, allowing us to grasp how often these outcomes happen together.

Picture a game where you roll two dice. The PMF helps you determine, for instance, the probability of rolling a 3 on the first die and a 4 on the second die. You can create a table listing all the possible outcomes, and this will help you see how likely each pair (like (3,4)) is compared to others.

3.2.2 Joint Probability Density Function (PDF)

For continuous random variables: 𝑓 (𝑥,𝑦) = joint PDF of 𝑋 and 𝑌

For continuous random variables, we use the Joint Probability Density Function (PDF). Unlike discrete variables, which have distinct probabilities, continuous variables can take on any value within an interval. The joint PDF, represented as f(x, y), describes how the probabilities of X and Y are distributed.

We can't directly calculate probabilities for specific values, like P(X = x, Y = y), because the probability of any single outcome is essentially zero. Instead, we look for the probability that X and Y fall within a specified range, which requires integrating the joint pdf over that range.

Imagine you are examining the time (X) a customer spends in a store and the amount of money (Y) they spend. The joint PDF describes how these two variables are likely to vary together. For example, you can't ask for the probability that a customer spends exactly $20 and stays exactly 15 minutes, but you can determine the likelihood of a customer spending between$15 and $25 while staying between 10 and 20 minutes, by looking at the area under the curve in the graph of the joint pdf.

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