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In probability theory, the marginal distribution is a significant concept that allows us to determine the distribution of a subset of variables within a larger set. Specifically, it tells us how a random variable behaves regardless of the other variables it may be connected with, hence "marginal". In the context of a joint probability mass function (pmf), the marginal distribution of a discrete random variable, like either of the variables \(X\) or \(Y\) in the given exercise, is obtained by summing over the probabilities of the other variables. This way, we can see the probability distribution of one random variable in isolation.For instance, to find the marginal pmf of \(X\), we sum the joint probabilities \(p(x, y)\) over all possible values of \(y\). Using the table in the exercise:

• \(p_X(0) = 0.10 + 0.04 + 0.02 = 0.16\)
• \(p_X(1) = 0.08 + 0.20 + 0.06 = 0.34\)
• \(p_X(2) = 0.06 + 0.14 + 0.30 = 0.50\)

Similarly, to find the marginal pmf of \(Y\), sum over all values of \(x\):

• \(p_Y(0) = 0.10 + 0.08 + 0.06 = 0.24\)
• \(p_Y(1) = 0.04 + 0.20 + 0.14 = 0.38\)
• \(p_Y(2) = 0.02 + 0.06 + 0.30 = 0.38\)

Marginal distributions provide critical insights into each variable's individual behavior, separating it from its interactions with other random variables.

When examining random variables for independence, we look for the condition where the occurrence of one does not affect the occurrence of the other. In mathematical terms, two random variables \(X\) and \(Y\) are independent if for all possible values of \(x\) and \(y\), the joint probability function can be expressed as the product of the two marginal probabilities: \[p(x, y) = p_X(x) \cdot p_Y(y)\]To test if \(X\) and \(Y\) are independent, we apply this condition to different combinations of \(x\) and \(y\). For example, in the exercise, check if:- For \((1,1)\): \(p(1,1) = 0.20\) and \(p_X(1)p_Y(1) = 0.34 \times 0.38 = 0.1292\) which are not equal.Since this condition does not hold true for at least one case, \(X\) and \(Y\) are not independent. This means the usage at one service island could affect the usage at the other.

Probability computation involves calculating the likelihood of various events occurring in a defined sample space. Whether it's finding a specific point probability or the cumulative probability of several events, a clear understanding of the rules and operations is essential.Take the question \(P(X\leq 1, Y\leq 1)\) as an example. This asks about the probability that each variable is less than or equal to one. To find this, you sum the probabilities of all combinations fitting this criterion. These combinations are \((0,0), (0,1), (1,0),\) and \((1,1)\), as computed in the step-by-step solution, producing:\[P(X \leq 1, Y \leq 1) = p(0,0) + p(0,1) + p(1,0) + p(1,1) = 0.10 + 0.04 + 0.08 + 0.20 = 0.42\]Correct computation often involves multiple steps:

• Identifying the relevant events.
• Summing or multiplying probabilities, depending on whether they are cumulative or independent.
• Checking if probabilities sum to 1 in the case of distribution functions for consistency.

These computations form the basis for making informed predictions and decisions based on probabilistic models.