In the context of probability, which of the following best describes the difference between a $\mathrm{marginal}$ distribution and a $\mathrm{conditional}$ distribution?

• A. A $\mathrm{marginal}$ distribution gives the probabilities of a single variable by summing or integrating over the possible values of other variables, while a $\mathrm{conditional}$ distribution gives the probabilities of one variable given that another variable has a specific value.
• B. A $\mathrm{marginal}$ distribution is only used for continuous variables, while a $\mathrm{conditional}$ distribution is only used for discrete variables.
• C. A $\mathrm{marginal}$ distribution gives the probabilities of two variables occurring together, while a $\mathrm{conditional}$ distribution gives the probability of either variable occurring independently.
• D. A $\mathrm{marginal}$ distribution and a $\mathrm{conditional}$ distribution are two names for the same concept in probability.

Answer: (A)

Verified step by step guidance

1. Step 1: Understand what a marginal distribution is. It represents the probabilities of a single variable by summing (for discrete variables) or integrating (for continuous variables) over the possible values of other variables in a joint distribution.
2. Step 2: Understand what a conditional distribution is. It represents the probabilities of one variable given that another variable has a specific value, essentially focusing on a subset of the data where the condition holds true.
3. Step 3: Recognize that marginal distributions provide an overall view of one variable without considering the influence of other variables, while conditional distributions show how the probability of one variable changes when another variable is fixed.
4. Step 4: Compare the given answer choices by matching these definitions: the correct description should highlight that marginal distributions summarize over other variables, and conditional distributions depend on a given value of another variable.
5. Step 5: Conclude that the correct answer is the one stating that a marginal distribution gives probabilities of a single variable by summing or integrating over other variables, and a conditional distribution gives probabilities of one variable given a specific value of another variable.