91Ó°ÊÓ

The Marginal Probability Mass Function (PMF) is a mathematical function that provides the probabilities of each possible value of a random variable. When dealing with joint distributions of two variables, say \(X\) and \(Y\), the marginal PMF represents the probabilities without considering the other variable. In our exercise, we have two random variables: \(X\), which denotes the number of headlights needing adjustment, and \(Y\), representing the number of defective tires.

• The marginal PMF for \(X\) is given by \(p_X(0) = 0.5\), \(p_X(1) = 0.3\), and \(p_X(2) = 0.2\).
• For \(Y\), it is \(p_Y(0) = 0.6\), \(p_Y(1) = 0.1\), \(p_Y(2) = 0.05\), \(p_Y(3) = 0.05\), and \(p_Y(4) = 0.2\).

Marginal PMFs are crucial as they allow us to analyze each variable individually, even though the variables may be part of a joint distribution. When you need to find specific probabilities involving only one of these variables, your lookup will be directly to these marginal PMFs.

Independence in Probability refers to the situation where the occurrence of one event does not affect the probability of the other event occurring. In the context of random variables, \(X\) and \(Y\) are independent if the distribution of one variable does not change when the other variable is known. In our exercise, the problem states that \(X\) and \(Y\) are independent.This means that for any possible values \(x\) and \(y\) that \(X\) and \(Y\) can take:\[p(X = x, Y = y) = p_X(x) \times p_Y(y)\]The beauty of independence in probability is evidenced here: you compute joint probabilities using individual distributions without worrying about their conditional dependencies. In real-world terms, fixing or not fixing a headlight has no impact on the number of defective tires found, and vice versa. This essential concept allows us to build the joint probability table directly from the marginal PMFs.

A Joint Probability Table is a grid that shows the probability of different combinations of outcomes for two or more random variables. In our exercise, we use this table to show the joint distribution of headlights needing adjustment (\(X\)) and defective tires (\(Y\)).Given the independence of \(X\) and \(Y\), the joint probability for any combination \((x, y)\) is obtained by multiplying the individual probabilities \(p_X(x)\) and \(p_Y(y)\). This results in a table where each cell represents the probability of a particular pair \((X = x, Y = y)\).The entries within the table are as follows:

• For \((X=0, Y=0)\), the probability is 0.30.
• For \((X=0, Y=1)\), it is 0.06.
• For \((X=1, Y=0)\), it equals 0.18, and so on.

By summing relevant probabilities from this table, you can answer various probability questions, such as finding \(P(X \leq 1, Y \leq 1)\) and \(P(X+Y \leq 1)\). The table is your central tool for quickly visualizing and accessing joint probabilities in a digestible format.