Joint Probability Mass Function Worksheets for University Statistics Students

📚 Topic Summary

The Joint Probability Mass Function (Joint PMF) describes the probability that two or more discrete random variables simultaneously take on specific values. Unlike single-variable PMFs, which deal with the probability of a single variable, the Joint PMF gives the probability of combinations of outcomes. Understanding Joint PMFs is crucial for analyzing relationships and dependencies between variables in statistics.

For two discrete random variables $X$ and $Y$, the Joint PMF is defined as $P(X = x, Y = y)$, representing the probability that $X$ takes the value $x$ and $Y$ takes the value $y$ at the same time. The sum of all probabilities across all possible pairs of values for $X$ and $Y$ must equal 1. This concept extends to more than two variables, but we'll focus on the two-variable case for simplicity.

🧮 Part A: Vocabulary

Match the term with its definition:

1. Term: Marginal Probability
2. Term: Joint Probability
3. Term: Random Variable
4. Term: Independence
5. Term: Conditional Probability

1. Definition: The probability of an event occurring given that another event has already occurred.
2. Definition: A variable whose value is a numerical outcome of a random phenomenon.
3. Definition: The probability of a single event occurring without regard to any other events.
4. Definition: The probability of two or more events occurring simultaneously.
5. Definition: Two events are independent if the occurrence of one does not affect the probability of the other.

📝 Part B: Fill in the Blanks

Complete the following paragraph with the correct words:

The Joint PMF, denoted as $P(X=x, Y=y)$, gives the ______ that two random variables, $X$ and $Y$, both take on specific values $x$ and $y$ __________. The sum of all joint probabilities over all possible pairs of values must equal _______. If $X$ and $Y$ are __________, then $P(X=x, Y=y) = P(X=x) * P(Y=y)$. In this case, they don't depend _________ on one another.

🤔 Part C: Critical Thinking

Explain, in your own words, why it is important to check that the sum of all probabilities in a Joint PMF equals 1. What would it mean if it didn't?