In the context of probability, which of the following best describes a $\mathrm{marginal}$ distribution as compared to a $\mathrm{conditional}$ distribution?

• A. A $\mathrm{marginal}$ distribution is only used for independent events, while a $\mathrm{conditional}$ distribution is used for dependent events.
• B. A $\mathrm{marginal}$ distribution gives the probabilities of one variable given another variable has a specific value, while a $\mathrm{conditional}$ distribution gives the probabilities of all variables together.
• C. 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.
• D. A $\mathrm{marginal}$ distribution and a $\mathrm{conditional}$ distribution are always the same for any two random variables.

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the concept of a marginal distribution. A marginal distribution provides the probabilities or probability density of a single random variable by summing (for discrete variables) or integrating (for continuous variables) over the possible values of other variables in the joint distribution. Mathematically, for two variables X and Y, the marginal distribution of X is given by: \[ P_X(x) = \sum_y P_{X,Y}(x,y) \] for discrete variables, or \[ f_X(x) = \int f_{X,Y}(x,y) \, dy \] for continuous variables.
2. Step 2: Understand the concept of a conditional distribution. A conditional distribution gives the probability distribution of one variable given that another variable takes a specific value. For two variables X and Y, the conditional distribution of X given Y = y is: \[ P_{X|Y}(x|y) = \frac{P_{X,Y}(x,y)}{P_Y(y)} \] for discrete variables, or \[ f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)} \] for continuous variables, where \(P_Y(y)\) or \(f_Y(y)\) is the marginal distribution of Y.
3. Step 3: Compare marginal and conditional distributions. The marginal distribution summarizes the probabilities of one variable without reference to the other variable's specific values, effectively 'averaging out' the other variable. The conditional distribution, on the other hand, focuses on the distribution of one variable when the other variable is fixed at a particular value.
4. Step 4: Clarify common misconceptions. Marginal distributions are not limited to independent events; they exist regardless of dependence. Conditional distributions are not the probabilities of all variables together but rather the distribution of one variable conditioned on another.
5. Step 5: Summarize the key difference. The marginal distribution is obtained by summing or integrating over the other variables to get the distribution of a single variable, while the conditional distribution is the distribution of one variable given a specific value of another variable.