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Joint probability distribution represents the probability that two random variables, say \(X\) and \(Y\), take on specific values. In the context of the multivariate hypergeometric distribution, it tells us how likely it is to select a certain number of students from industrial and mechanical engineering majors from the sample. In this exercise, we use a combination formula to determine this probability. The joint probability distribution function is defined as:\[ f_{XY}(x, y) = \frac{{\binom{4}{x} \binom{12}{y} \binom{24}{(4-x-y)}}}{{\binom{40}{4}}} \]- \(\binom{n}{k}\) represents the number of ways to choose \(k\) elements from \(n\) elements.- This formula accounts for all ways of picking \(x\) industrial majors, \(y\) mechanical majors, and the rest as electrical majors without exceeding the total sample size of 4.This probability spread gives a comprehensive look at all possible combinations.

The marginal distribution is like zooming out from the joint distribution, focusing only on one of the variables (e.g., \(X\) or \(Y\)) by summing over the probabilities of the other variable.For \(X\), the number of industrial engineering majors, the marginal distribution is found by:\[ f_X(x) = \sum_{y=0}^{4-x} f_{XY}(x, y) \]This approach aggregates the probability over all possible values of \(y\) that satisfy the sample condition. Marginal distributions help to understand the behavior of one variable independently by collapsing the joint distribution.

Expected value is a fundamental concept in probability representing the average or mean value that a random variable takes on. It is like a long-term average outcome.To find the expected value of \(X\), \(E(X)\), use the formula:\[ E(X) = \sum_{x=0}^{4} x \cdot f_X(x) \]This takes into account each value of \(x\), weighted by its marginal probability. Expected values provide insights into the most likely outcomes over repeated sampling and serve as important summaries for random variables.

Conditional distribution becomes relevant when one random variable's value is known. It's about recalculating probabilities given this new information.For \(Y\) given \(X=3\), the conditional distribution is determined by:\[ f_{Y \mid 3}(y) = \frac{f_{XY}(3, y)}{f_X(3)} \]This isolates the probability of \(Y\) when \(X\) has already taken a specific value, here 3. Conditional distribution is pivotal for any scenario where one outcome affects the chances of others, helping to reassess situations based on known data.

Variance quantifies the spread of a random variable’s possible values, indicating how much the values deviate from the mean. A higher variance reflects more spread out data.Calculating the variance of \(Y\) given \(X=3\) involves two steps:1. Calculate \(E(Y^2 \mid X=3)\):\[ E(Y^2 \mid X=3) = \sum_{y=0}^{1} y^2 \cdot f_{Y \mid 3}(y) \]2. Use the variance formula:\[ V(Y \mid X=3) = E(Y^2 \mid X=3) - (E(Y \mid X=3))^2 \]This helps to understand the variability of \(Y\) under a specific condition on \(X\), aiding in predicting the consistency of outcomes.

Independence between two random variables, \(X\) and \(Y\), indicates that the occurrence of one does not affect the occurrence of the other.Two variables are independent if:\[ f_{XY}(x, y) = f_X(x) \cdot f_Y(y) \]However, with students selected concurrently without replacement, \(X\) and \(Y\) are not independent in this scenario. Selection affects the remaining choices, intertwining their probabilities. Recognizing dependencies is essential in correctly modeling the joint effects of variables.