Calculating Marginal Probability from Joint PMF

For a given joint PMF of three random variables X, Y, and Z, determine the probability that X takes on a specific value. Consider all possible triples where random variable X indeed takes that value and sum over all possible values of Y's and Z's that go together with this particular X.

In this problem, you are asked to find the probability that a random variable X takes on a specific value, given a joint probability mass function (PMF) involving three random variables: X, Y, and Z. The key concept here is the idea of marginal probability, which refers to the probability of a single event occurring irrespective of the outcomes of related random variables. To solve this, you will sum over the joint probabilities of all possible outcomes for Y and Z that result in the desired value of X.

This approach will provide the marginal probability of X for the specified value. Marginal probabilities derive from the broader concept of joint distributions, which encapsulate the probabilities of different combinations of outcomes for multiple random variables. Understanding how to extract marginal probabilities is fundamental as it allows you to simplify complex problems involving multiple random variables by reducing them to single-variable scenarios.

This technique is crucial in many fields, such as economics, engineering, and social sciences, where interpreting data from multivariate contexts is a common task.

Additionally, this problem implicitly touches on the concept of independence between random variables. If X, Y, and Z were independent, finding the probability of X would not require summing over Y and Z. Grasping this distinction is key in probability and statistics, especially when developing models or analyzing real-world data where independence assumptions can significantly simplify calculations.