In the context of probability, which of the following best describes a marginal distribution as compared to a conditional distribution?

• A. A marginal distribution is always uniform, while a conditional distribution is always normal.
• B. A marginal distribution gives the probabilities of one variable only when another variable is fixed at a certain value, while a conditional distribution gives the probabilities of all variables together.
• C. A marginal distribution is the probability of the $\cap$ (intersection) of two events, while a conditional distribution is the probability of the $\cup$ (union) of two events.
• D. A marginal distribution gives the probabilities of a single variable by $\sum$ (summing) or $\int$ (integrating) over the possible values of the other variable(s), while a conditional distribution gives the probabilities of one variable given that another variable has a specific value (i.e., $P\left(A\right|B)$).

Answer: (D)

Verified step by step guidance

1. Understand that a marginal distribution focuses on the probabilities of a single variable without considering the specific values of other variables. It is obtained by summing (for discrete variables) or integrating (for continuous variables) the joint distribution over the other variables.
2. Recognize that a conditional distribution describes the probability of one variable given that another variable is fixed at a certain value. This means it looks at the distribution of one variable under the condition that the other variable is known.
3. Recall the formula for marginal distribution from a joint distribution \(P(X, Y)\): the marginal distribution of \(X\) is given by \(P(X = x) = \sum_y P(X = x, Y = y)\) for discrete variables, or \(P(X = x) = \int P(X = x, Y = y) \, dy\) for continuous variables.
4. Recall the formula for conditional distribution: \(P(X = x \mid Y = y) = \frac{P(X = x, Y = y)}{P(Y = y)}\), which shows how the probability of \(X\) changes when \(Y\) is fixed at a particular value.
5. Compare the two concepts: marginal distribution aggregates over the other variable(s) to focus on one variable alone, while conditional distribution restricts the focus to a subset of the sample space where the other variable(s) have specific values.