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A joint probability distribution gives us the probability of two random variables occurring simultaneously. In this exercise, we have two variables: the number of married executives, \(Y_1\), and the number of never-married executives, \(Y_2\). The joint probability distribution is represented by \(p(y_1, y_2)\), which provides probabilities for different combinations of \(y_1\) and \(y_2\).
The formula provided in the exercise is\[p\left(y_{1}, y_{2}\right)=\frac{\binom{4}{y_1}\binom{3}{y_2}\binom{2}{3-y_1-y_2}}{\binom{9}{3}}\]Here:

• \(\binom{n}{k}\) is the binomial coefficient, equal to \(\frac{n!}{k!(n-k)!}\).
• \(9\) is the total number of executives, while \(3\) is the sum of executives being considered for marriage status.

The probability is essentially calculated by considering how these individuals can be grouped based on their marital status. Each valid \(y_1, y_2\) pair has an associated probability that reflects the likely occurrence of that combination. This distribution is essential for understanding how the number of married and never-married executives relate to each other probabilistically.

The expected value is a fundamental concept in probability, often referred to as the mean or the average. When dealing with random variables, it tells us the average outcome we can expect over a large number of trials. For a random variable \(Y\), the expected value \(E(Y)\) can be calculated by summing up all possible values it can take, each multiplied by its probability.
In the case of our exercise:- To find \(E(Y_1)\), you compute \(\sum_{y_1, y_2} y_1 \cdot p(y_1, y_2)\). This means you multiply each value \(y_1\) by its associated joint probability and sum them up across all combinations of \(y_1\) and \(y_2\).- Similarly, \(E(Y_2) = \sum_{y_1, y_2} y_2 \cdot p(y_1, y_2)\) follows the same process, focusing on \(y_2\).
Ultimately, the expected values \(E(Y_1)\) and \(E(Y_2)\) give an average sense of how many married versus never-married executives we can expect, based on our joint probability distribution.

Covariance is a measure of how much two random variables change together. If two variables tend to increase together, the covariance is positive. If one tends to decrease when the other increases, it's negative. The formula for covariance \(\operatorname{Cov}(Y_1, Y_2)\) is given by:\[\operatorname{Cov}(Y_1, Y_2) = E(Y_1 Y_2) - E(Y_1)E(Y_2)\]To find \(E(Y_1 Y_2)\), we calculate \(\sum_{y_1, y_2} y_1 y_2 \cdot p(y_1, y_2)\), where each product \(y_1 y_2\) is weighted by the probability \(p(y_1, y_2)\) and summed over all possibilities.
This outcome \(E(Y_1 Y_2)\) represents the expected value of the product of \(Y_1\) and \(Y_2\). By subtracting the product of \(E(Y_1)\) and \(E(Y_2)\), we can see how the actual joint expectation differs from what we would expect if \(Y_1\) and \(Y_2\) were independent.
Thus, covariance helps us understand the relationship between how often executives are married or never-married, revealing whether an increase in one correlates with a decrease or increase in the other.